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Introductory Chemical Engineering Thermodynamics, Second Edition Prentice Hall International Series in the Physical and Chemical Engineering Sciences Visit informit.com/ph/physandchem for a complete list of available publications. The Prentice Hall International Series in the Physical and Chemical Engineering Sciences had its auspicious beginning in 1956 under the direction of Neal R. Amundsen. The series comprises the most widely adopted college textbooks and supplements for chemical engineering education. Books in this series are written by the foremost educators and researchers in the field of chemical engineering. Make sure to connect with us! informit.com/socialconnect Introductory Chemical Engineering Thermodynamics, Second Edition J. Richard Elliott Carl T. 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First printing, February 2012 CONTENTS PREFACE Notes to Students Acknowledgments xvii xviii xviii ABOUT THE AUTHORS xix GLOSSARY xxi NOTATION xxv UNIT I FIRST AND SECOND LAWS 1 CHAPTER 1 BASIC CONCEPTS 3 1.1 Introduction 1.2 The Molecular Nature of Energy, Temperature, and Pressure Example 1.1 The energy derived from intermolecular potentials Example 1.2 Intermolecular potentials for mixtures 1.3 The Molecular Nature of Entropy 1.4 Basic Concepts 1.5 Real Fluids and Tabulated Properties Example 1.3 Introduction to steam tables Example 1.4 Interpolation Example 1.5 Double interpolation Example 1.6 Double interpolation using different tables Example 1.7 Double interpolation using Excel Example 1.8 Quality calculations Example 1.9 Constant volume cooling 1.6 Summary 1.7 Practice Problems 5 6 12 14 15 15 22 25 27 27 28 29 31 32 33 34 v vi Contents 1.8 Homework Problems CHAPTER 2 THE ENERGY BALANCE 2.1 2.2 2.3 2.4 Expansion/Contraction Work Shaft Work Work Associated with Flow Lost Work versus Reversibility Example 2.1 Isothermal reversible compression of an ideal gas 2.5 Heat Flow 2.6 Path Properties and State Properties Example 2.2 Work as a path function 2.7 The Closed-System Energy Balance Example 2.3 Internal energy and heat 2.8 The Open-System, Steady-State Balance Example 2.4 Pump work for compressing H2O 2.9 The Complete Energy Balance 2.10 Internal Energy, Enthalpy, and Heat Capacities Example 2.5 Enthalpy change of an ideal gas: Integrating CPig(T) Example 2.6 Enthalpy of compressed liquid Example 2.7 Adiabatic compression of an ideal gas in a piston/cylinder 2.11 Reference States Example 2.8 Acetone enthalpy using various reference states 2.12 Kinetic and Potential Energy Example 2.9 Comparing changes in kinetic energy, potential energy, internal energy, and enthalpy Example 2.10 Transformation of kinetic energy into enthalpy 2.13 Energy Balances for Process Equipment 2.14 Strategies for Solving Process Thermodynamics Problems 2.15 Closed and Steady-State Open Systems Example 2.11 Adiabatic, reversible expansion of an ideal gas Example 2.12 Continuous adiabatic, reversible compression of an ideal gas Example 2.13 Continuous, isothermal, reversible compression of an ideal gas Example 2.14 Heat loss from a turbine 2.16 Unsteady-State Open Systems Example 2.15 Adiabatic expansion of an ideal gas from a leaky tank Example 2.16 Adiabatically filling a tank with an ideal gas Example 2.17 Adiabatic expansion of steam from a leaky tank 2.17 Details of Terms in the Energy Balance 2.18 Summary 2.19 Practice Problems 2.20 Homework Problems CHAPTER 3 ENERGY BALANCES FOR COMPOSITE SYSTEMS 3.1 Heat Engines and Heat Pumps – The Carnot Cycle Example 3.1 Analyzing heat pumps for housing 3.2 Distillation Columns Example 3.2 Start-up for a distillation column 3.3 Introduction to Mixture Properties 3.4 Ideal Gas Mixture Properties 35 39 40 41 41 42 45 46 46 46 48 50 51 55 56 57 60 60 61 63 65 66 66 67 68 74 75 75 76 78 79 80 81 82 83 85 86 88 90 95 96 100 101 103 105 106 Contents 3.5 Mixture Properties for Ideal Solutions Example 3.3 Condensation of a vapor stream 3.6 Energy Balance for Reacting Systems Example 3.4 Stoichiometry and the reaction coordinate Example 3.5 Using the reaction coordinates for simultaneous reactions Example 3.6 Reactor energy balances 3.7 Reactions in Biological Systems 3.8 Summary 3.9 Practice Problems 3.10 Homework Problems CHAPTER 4 ENTROPY 4.1 The Concept of Entropy 4.2 The Microscopic View of Entropy Example 4.1 Entropy change and “lost work” in a gas expansion Example 4.2 Stirling’s approximation in the Einstein solid 4.3 The Macroscopic View of Entropy Example 4.3 Adiabatic, reversible expansion of steam Example 4.4 A Carnot cycle based on steam Example 4.5 Ideal gas entropy changes in an adiabatic, reversible expansion Example 4.6 Ideal gas entropy change: Integrating CPig(T) Example 4.7 Entropy generation and “lost work” Example 4.8 Entropy generation in a temperature gradient 4.4 The Entropy Balance Example 4.9 Entropy balances for steady-state composite systems 4.5 Internal Reversibility 4.6 Entropy Balances for Process Equipment Example 4.10 Entropy generation by quenching Example 4.11 Entropy in a heat exchanger Example 4.12 Isentropic expansion in a nozzle 4.7 Turbine, Compressor, and Pump Efficiency 4.8 Visualizing Energy and Entropy Changes 4.9 Turbine Calculations Example 4.13 Various cases of turbine outlet conditions Example 4.14 Turbine efficiency calculation Example 4.15 Turbine inlet calculation given efficiency and outlet 4.10 Pumps and Compressors Example 4.16 Isothermal reversible compression of steam Example 4.17 Compression of R134a using P-H chart 4.11 Strategies for Applying the Entropy Balance 4.12 Optimum Work and Heat Transfer Example 4.18 Minimum heat and work of purification 4.13 The Irreversibility of Biological Life 4.14 Unsteady-State Open Systems Example 4.19 Entropy change in a leaky tank Example 4.20 An ideal gas leaking through a turbine (unsteady state) 4.15 The Entropy Balance in Brief 4.16 Summary 4.17 Practice Problems vii 106 107 109 110 111 116 119 121 122 122 129 130 132 137 141 142 144 145 149 151 151 152 153 155 158 159 159 160 162 164 165 166 168 171 172 173 173 174 175 177 180 181 182 182 183 185 185 187 viii Contents 4.18 Homework Problems CHAPTER 5 THERMODYNAMICS OF PROCESSES 5.1 The Carnot Steam Cycle 5.2 The Rankine Cycle Example 5.1 Rankine cycle 5.3 Rankine Modifications Example 5.2 A Rankine cycle with reheat Example 5.3 Regenerative Rankine cycle 5.4 Refrigeration Example 5.4 Refrigeration by vapor compression cycle 5.5 Liquefaction Example 5.5 Liquefaction of methane by the Linde process 5.6 Engines 5.7 Fluid Flow 5.8 Problem-Solving Strategies 5.9 Summary 5.10 Practice Problems 5.11 Homework Problems 189 199 199 200 201 203 204 206 208 209 212 213 214 214 214 215 215 216 UNIT II GENERALIZED ANALYSIS OF FLUID PROPERTIES 223 CHAPTER 6 CLASSICAL THERMODYNAMICS — GENERALIZATIONS FOR ANY FLUID 225 6.1 The Fundamental Property Relation 6.2 Derivative Relations Example 6.1 Pressure dependence of H Example 6.2 Entropy change with respect to T at constant P Example 6.3 Entropy as a function of T and P Example 6.4 Entropy change for an ideal gas Example 6.5 Entropy change for a simple nonideal gas Example 6.6 Accounting for T and V impacts on energy Example 6.7 The relation between Helmholtz energy and internal energy Example 6.8 A quantum explanation of low T heat capacity Example 6.9 Volumetric dependence of CV for ideal gas Example 6.10 Application of the triple product relation Example 6.11 Master equation for an ideal gas Example 6.12 Relating CP to CV 6.3 Advanced Topics 6.4 Summary 6.5 Practice Problems 6.6 Homework Problems 226 229 233 234 235 237 237 238 239 240 242 243 243 244 244 247 248 248 CHAPTER 7 ENGINEERING EQUATIONS OF STATE FOR PVT PROPERTIES 251 7.1 Experimental Measurements 7.2 Three-Parameter Corresponding States 7.3 Generalized Compressibility Factor Charts 252 253 256 Contents Example 7.1 Application of the generalized charts 7.4 The Virial Equation of State Example 7.2 Application of the virial equation 7.5 Cubic Equations of State 7.6 Solving the Cubic Equation of State for Z Example 7.3 Peng-Robinson solution by hand calculation Example 7.4 The Peng-Robinson equation for molar volume Example 7.5 Application of the Peng-Robinson equation 7.7 Implications of Real Fluid Behavior Example 7.6 Derivatives of the Peng-Robinson equation 7.8 Matching the Critical Point Example 7.7 Critical parameters for the van der Waals equation 7.9 The Molecular Basis of Equations of State: Concepts and Notation Example 7.8 Estimating molecular size Example 7.9 Characterizing molecular interactions 7.10 The Molecular Basis of Equations of State: Molecular Simulation Example 7.10 Computing molecular collisions in 2D Example 7.11 Equations of state from trends in molecular simulations 7.11 The Molecular Basis of Equations of State: Analytical Theories Example 7.12 Deriving your own equation of state 7.12 Summary 7.13 Practice Problems 7.14 Homework Problems CHAPTER 8 DEPARTURE FUNCTIONS 8.1 The Departure Function Pathway 8.2 Internal Energy Departure Function Example 8.1 Internal energy departure from the van der Waals equation 8.3 Entropy Departure Function 8.4 Other Departure Functions 8.5 Summary of Density-Dependent Formulas 8.6 Pressure-Dependent Formulas 8.7 Implementation of Departure Formulas Example 8.2 Real entropy in a combustion engine Example 8.3 Compression of methane using the virial equation Example 8.4 Computing enthalpy and entropy departures from the Peng-Robinson equation Example 8.5 Enthalpy departure for the Peng-Robinson equation Example 8.6 Gibbs departure for the Peng-Robinson equation Example 8.7 U and S departure for the Peng-Robinson equation 8.8 Reference States Example 8.8 Enthalpy and entropy from the Peng-Robinson equation Example 8.9 Liquefaction revisited Example 8.10 Adiabatically filling a tank with propane 8.9 Generalized Charts for the Enthalpy Departure 8.10 Summary 8.11 Practice Problems 8.12 Homework Problems ix 258 258 259 260 263 266 266 268 269 269 270 271 271 273 275 276 279 281 282 288 289 290 291 301 302 304 306 307 308 308 309 310 310 312 314 316 317 317 318 320 320 322 323 323 325 326 x Contents CHAPTER 9 PHASE EQUILIBRIUM IN A PURE FLUID 9.1 Criteria for Phase Equilibrium 9.2 The Clausius-Clapeyron Equation Example 9.1 Clausius-Clapeyron equation near or below the boiling point 9.3 Shortcut Estimation of Saturation Properties Example 9.2 Vapor pressure interpolation Example 9.3 Application of the shortcut vapor pressure equation Example 9.4 General application of the Clapeyron equation 9.4 Changes in Gibbs Energy with Pressure 9.5 Fugacity and Fugacity Coefficient 9.6 Fugacity Criteria for Phase Equilibria 9.7 Calculation of Fugacity (Gases) 9.8 Calculation of Fugacity (Liquids) Example 9.5 Vapor and liquid fugacities using the virial equation 9.9 Calculation of Fugacity (Solids) 9.10 Saturation Conditions from an Equation of State Example 9.6 Vapor pressure from the Peng-Robinson equation Example 9.7 Acentric factor for the van der Waals equation Example 9.8 Vapor pressure using equal area rule 9.11 Stable Roots and Saturation Conditions 9.12 Temperature Effects on G and f 9.13 Summary 9.14 Practice Problems 9.15 Homework Problems UNIT III FLUID PHASE EQUILIBRIA IN MIXTURES 335 336 337 338 339 339 341 341 342 344 346 347 348 352 353 353 354 356 359 359 361 361 362 363 367 CHAPTER 10 INTRODUCTION TO MULTICOMPONENT SYSTEMS 369 10.1 10.2 10.3 10.4 Introduction to Phase Diagrams Vapor-Liquid Equilibrium (VLE) Calculations Binary VLE Using Raoult’s Law Multicomponent VLE Raoult’s Law Calculations Example 10.1 Bubble and dew temperatures and isothermal flash of ideal solutions Example 10.2 Adiabatic flash 10.5 Emissions and Safety 10.6 Relating VLE to Distillation 10.7 Nonideal Systems 10.8 Concepts for Generalized Phase Equilibria 10.9 Mixture Properties for Ideal Gases 10.10 Mixture Properties for Ideal Solutions 10.11 The Ideal Solution Approximation and Raoult’s Law 10.12 Activity Coefficient and Fugacity Coefficient Approaches 10.13 Summary 10.14 Practice Problems 10.15 Homework Problems CHAPTER 11 AN INTRODUCTION TO ACTIVITY MODELS 11.1 Modified Raoult’s Law and Excess Gibbs Energy 370 372 374 381 382 385 386 390 393 397 401 403 404 405 405 407 407 411 412 Contents Example 11.1 Gibbs excess energy for system 2-propanol water 11.2 Calculations Using Activity Coefficients Example 11.2 VLE predictions from the Margules equation Example 11.3 Gibbs excess characterization by matching the bubble point Example 11.4 Predicting the Margules parameter with the MAB model 11.3 Deriving Modified Raoult’s Law 11.4 Excess Properties 11.5 Modified Raoult’s Law and Excess Gibbs Energy 11.6 Redlich-Kister and the Two-Parameter Margules Models Example 11.5 Fitting one measurement with the two-parameter Margules equation Example 11.6 Dew pressure using the two-parameter Margules equation 11.7 Activity Coefficients at Special Compositions Example 11.7 Azeotrope fitting with bubble-temperature calculations 11.8 Preliminary Indications of VLLE 11.9 Fitting Activity Models to Multiple Data Example 11.8 Fitting parameters using nonlinear least squares 11.10 Relations for Partial Molar Properties Example 11.9 Heats of mixing with the Margules two-parameter model 11.11 Distillation and Relative Volatility of Nonideal Solutions Example 11.10 Suspecting an azeotrope 11.12 Lewis-Randall Rule and Henry’s Law Example 11.11 Solubility of CO2 by Henry’s Law Example 11.12 Henry’s constant for CO2 with the MAB/SCVP+ model 11.13 Osmotic Pressure Example 11.13 Osmotic pressure of BSA Example 11.14 Osmotic pressure and electroporation of E. coli 11.14 Summary 11.15 Practice Problems 11.16 Homework Problems CHAPTER 12 VAN DER WAALS ACTIVITY MODELS 12.1 The van der Waals Perspective for Mixtures 12.2 The van Laar Model Example 12.1 Infinite dilution activity coefficients from the van Laar theory 12.3 Scatchard-Hildebrand Theory Example 12.2 VLE predictions using the Scatchard-Hildebrand theory 12.4 The Flory-Huggins Model Example 12.3 Deriving activity models involving volume fractions Example 12.4 Scatchard-Hildebrand versus van Laar theory for methanol benzene Example 12.5 Polymer mixing 12.5 MOSCED and SSCED Theories Example 12.6 Predicting VLE with the SSCED model 12.6 Molecular Perspective and VLE Predictions 12.7 Multicomponent Extensions of van der Waals’ Models Example 12.7 Multicomponent VLE using the SSCED model Example 12.8 Entrainer selection for gasohol production 12.8 Flory-Huggins and van der Waals Theories 12.9 Summary 12.10 Practice Problems xi 414 416 417 418 423 423 426 427 429 430 431 432 433 434 435 436 439 441 442 442 443 446 448 449 451 453 454 455 455 465 466 469 471 471 473 474 475 476 478 479 482 483 486 487 490 491 492 494 xii Contents 12.11 Homework Problems 495 CHAPTER 13 LOCAL COMPOSITION ACTIVITY MODELS 499 Example 13.1 VLE prediction using UNIFAC activity coefficients 13.1 Local Composition Theory Example 13.2 Local compositions in a two-dimensional lattice 13.2 Wilson’s Equation Example 13.3 Application of Wilson’s equation to VLE 13.3 NRTL 13.4 UNIQUAC Example 13.4 Combinatorial contribution to the activity coefficient 13.5 UNIFAC Example 13.5 Calculation of group mole fractions Example 13.6 Detailed calculations of activity coefficients via UNIFAC 13.6 COSMO-RS Methods Example 13.7 Calculation of activity coefficients using COSMO-RS/SAC 13.7 The Molecular Basis of Solution Models 13.8 Summary 13.9 Important Equations 13.10 Practice Problems 13.11 Homework Problems 500 501 503 505 508 508 509 512 514 518 518 520 524 526 532 533 533 534 CHAPTER 14 LIQUID-LIQUID AND SOLID-LIQUID PHASE EQUILIBRIA 14.1 The Onset of Liquid-Liquid Instability Example 14.1 Simple vapor-liquid-liquid equilibrium (VLLE) calculations Example 14.2 LLE predictions using Flory-Huggins theory: Polymer mixing 14.2 Stability and Excess Gibbs Energy 14.3 Binary LLE by Graphing the Gibbs Energy of Mixing Example 14.3 LLE predictions by graphing 14.4 LLE Using Activities Example 14.4 The binary LLE algorithm using MAB and SSCED models 14.5 VLLE with Immiscible Components Example 14.5 Steam distillation 14.6 Binary Phase Diagrams 14.7 Plotting Ternary LLE Data 14.8 Critical Points in Binary Liquid Mixtures Example 14.6 Liquid-liquid critical point of the Margules one-parameter model Example 14.7 Liquid-liquid critical point of the Flory-Huggins model 14.9 Numerical Procedures for Binary, Ternary LLE 14.10 Solid-Liquid Equilibria Example 14.8 Variation of solid solubility with temperature Example 14.9 Eutectic behavior of chloronitrobenzenes Example 14.10 Eutectic behavior of benzene phenol Example 14.11 Precipitation by adding antisolvent Example 14.12 Wax precipitation 14.11 Summary 14.12 Practice Problems 14.13 Homework Problems 539 539 540 541 542 543 544 545 547 548 548 549 551 552 553 554 556 556 560 561 562 563 567 569 570 570 Contents CHAPTER 15 PHASE EQUILIBRIA IN MIXTURES BY AN EQUATION OF STATE 15.1 Mixing Rules for Equations of State Example 15.1 The virial equation for vapor mixtures 15.2 Fugacity and Chemical Potential from an EOS Example 15.2 K-values from the Peng-Robinson equation 15.3 Differentiation of Mixing Rules Example 15.3 Fugacity coefficient from the virial equation Example 15.4 Fugacity coefficient from the van der Waals equation Example 15.5 Fugacity coefficient from the Peng-Robinson equation 15.4 VLE Calculations by an Equation of State Example 15.6 Bubble-point pressure from the Peng-Robinson equation Example 15.7 Isothermal flash using the Peng-Robinson equation Example 15.8 Phase diagram for azeotropic methanol benzene Example 15.9 Phase diagram for nitrogen + methane Example 15.10 Ethane + heptane phase envelopes 15.5 Strategies for Applying VLE Routines 15.6 Summary 15.7 Practice Problems 15.8 Homework Problems CHAPTER 16 ADVANCED PHASE DIAGRAMS 16.1 16.2 16.3 16.4 16.5 Phase Behavior Sections of 3D Objects Classification of Binary Phase Behavior Residue Curves Practice Problems Homework Problems xiii 579 580 581 582 587 588 590 591 592 594 594 596 598 599 601 603 603 604 606 613 613 617 630 636 636 UNIT IV REACTION EQUILIBRIA 639 CHAPTER 17 REACTION EQUILIBRIA 641 17.1 Introduction Example 17.1 Computing the reaction coordinate 17.2 Reaction Equilibrium Constraint 17.3 The Equilibrium Constant 17.4 The Standard State Gibbs Energy of Reaction Example 17.2 Calculation of standard state Gibbs energy of reaction 17.5 Effects of Pressure, Inerts, and Feed Ratios Example 17.3 Butadiene production in the presence of inerts 17.6 Determining the Spontaneity of Reactions 17.7 Temperature Dependence of Ka Example 17.4 Equilibrium constant as a function of temperature 17.8 Shortcut Estimation of Temperature Effects Example 17.5 Application of the shortcut van’t Hoff equation 17.9 Visualizing Multiple Equilibrium Constants 17.10 Solving Equilibria for Multiple Reactions Example 17.6 Simultaneous reactions that can be solved by hand Example 17.7 Solving multireaction equilibria with Excel 642 643 644 646 647 648 649 650 652 652 653 655 656 656 658 658 660 xiv Contents 17.11 Driving Reactions by Chemical Coupling Example 17.8 Chemical coupling to induce conversion 17.12 Energy Balances for Reactions Example 17.9 Adiabatic reaction in an ammonia reactor 17.13 Liquid Components in Reactions Example 17.10 Oligomerization of lactic acid 17.14 Solid Components in Reactions Example 17.11 Thermal decomposition of methane 17.15 Rate Perspectives in Reaction Equilibria 17.16 Entropy Generation via Reactions 17.17 Gibbs Minimization Example 17.12 Butadiene by Gibbs minimization Example 17.13 Direct minimization of the Gibbs energy with Excel Example 17.14 Pressure effects for Gibbs energy minimization 17.18 Reaction Modeling with Limited Data 17.19 Simultaneous Reaction and VLE Example 17.15 The solvent methanol process Example 17.16 NO2 absorption 17.20 Summary 17.21 Practice Problems 17.22 Homework Problems CHAPTER 18 ELECTROLYTE SOLUTIONS 18.1 Introduction to Electrolyte Solutions 18.2 Colligative Properties Example 18.1 Freezing point depression Example 18.2 Example of osmotic pressure Example 18.3 Example of boiling point elevation 18.3 Speciation and the Dissociation Constant 18.4 Concentration Scales and Standard States 18.5 The Definition of pH 18.6 Thermodynamic Network for Electrolyte Equilibria 18.7 Perspectives on Speciation 18.8 Acids and Bases Example 18.4 Dissociation of fluconazole 18.9 Sillèn Diagram Solution Method Example 18.5 Sillèn diagram for HOAc and NaOAc Example 18.6 Phosphate salt and strong acid Example 18.7 Distribution of species in glycine solution 18.10 Applications Example 18.8 Dissociation and solubility of fluconazole 18.11 Redox Reactions Example 18.9 Alkaline dry-cell battery 18.12 Biological Reactions Example 18.10 ATP hydrolysis Example 18.11 Biological fuel cell 18.13 Nonideal Electrolyte Solutions: Background 18.14 Overview of Model Development 18.15 The Extended Debye-Hückel Activity Model 662 663 664 665 667 668 669 670 671 672 673 673 675 676 677 677 677 681 683 684 686 693 693 695 695 696 697 697 699 701 702 703 704 709 712 713 717 721 723 726 727 730 731 737 738 739 740 742 Contents xv 18.16 Gibbs Energies for Electrolytes 18.17 Transformed Biological Gibbs Energies and Apparent Equilibrium Constants Example 18.12 Gibbs energy of formation for ATP 18.18 Coupled Multireaction and Phase Equilibria Example 18.13 Chlorine water electrolyte solutions 18.19 Mean Ionic Activity Coefficients 18.20 Extending Activity Calculations to High Concentrations 18.21 Summary 18.22 Supplement 1: Interconversion of Concentration Scales 18.23 Supplement 2: Relation of Apparent Chemical Potential to Species Potentials 18.24 Supplement 3: Standard States 18.25 Supplement 4: Conversion of Equilibrium Constants 18.26 Practice Problems 18.27 Homework Problems 743 745 748 749 750 753 755 755 757 758 759 760 761 761 CHAPTER 19 MOLECULAR ASSOCIATION AND SOLVATION 767 19.1 19.2 19.3 19.4 Introducing the Chemical Contribution Equilibrium Criteria Balance Equations for Binary Systems Ideal Chemical Theory for Binary Systems Example 19.1 Compressibility factors in associating/solvating systems Example 19.2 Dimerization of carboxylic acids Example 19.3 Activity coefficients in a solvated system 19.5 Chemical-Physical Theory 19.6 Wertheim’s Theory for Complex Mixtures Example 19.4 The chemical contribution to the equation of state 19.7 Mass Balances for Chain Association Example 19.5 Molecules of H2O in a 100 ml beaker 19.8 The Chemical Contribution to the Fugacity Coefficient and Compressibility Factor 19.9 Wertheim’s Theory of Polymerization Example 19.6 Complex fugacity for the van der Waals model Example 19.7 More complex fugacity for the van der Waals model 19.10 Statistical Associating Fluid Theory (The SAFT Model) Example 19.8 The SAFT model 19.11 Fitting the Constants for an Associating Equation of State 19.12 Summary 19.13 Practice Problems 19.14 Homework Problems APPENDIX A SUMMARY OF COMPUTER PROGRAMS A.1 A.2 A.3 A.4 A.5 A.6 Programs for Pure Component Properties Programs for Mixture Phase Equilibria Reaction Equilibria Notes on Excel Spreadsheets Notes on MATLAB Disclaimer 768 772 775 776 777 778 779 779 782 788 792 793 793 795 796 798 799 800 802 804 806 806 811 811 812 813 813 814 815 xvi Contents APPENDIX B MATHEMATICS B.1 Important Relations B.2 Solutions to Cubic Equations B.3 The Dirac Delta Function Example B.1 The hard-sphere equation of state Example B.2 The square-well equation of state APPENDIX C STRATEGIES FOR SOLVING VLE PROBLEMS C.1 Modified Raoult’s Law Methods C.2 EOS Methods C.3 Activity Coefficient (Gamma-Phi) Methods APPENDIX D MODELS FOR PROCESS SIMULATORS D.1 D.2 D.3 D.4 D.5 Overview Equations of State Solution Models Hybrid Models Recommended Decision Tree APPENDIX E THEMODYNAMIC PROPERTIES E.1 Thermochemical Data E.2 Latent Heats E.3 Antoine Constants E.4 Henry’s Constant with Water as Solvent E.5 Dielectric Constant for Water E.6 Dissociation Constants of Polyprotic Acids E.7 Standard Reduction Potentials E.8 Biochemical Data E.9 Properties of Water E.10 Pressure-Enthalpy Diagram for Methane E.11 Pressure-Enthalpy Diagram for Propane E.12 Pressure-Enthalpy Diagram for R134a (1,1,1,2-Tetraflouroethane) INDEX 817 817 822 825 828 829 831 832 835 838 839 839 839 840 840 841 843 843 846 847 847 848 849 849 852 854 865 866 867 869 PREFACE “No happy phrase of ours is ever quite original with us; there is nothing of our own in it except some slight change born of our temperament, character, environment, teachings and associations.” Mark Twain This textbook is designed for chemical engineering students from the sophomore level to the first year of graduate school. The approach blends molecular perspective with principles of thermodynamics to build intuitive reasoning regarding the behavior of species in chemical engineering processes and formulations. The molecular perspective is represented by descriptions encompassing: the relation of kinetic energy to temperature; the origin and consequences of intermolecular potentials; molecular acidity and basicity; methods used to incorporate molecular properties into molecular simulations; and the impact of molecular properties on macroscopic energy and entropy. This text is distinctive in making molecular perspectives accessible at the introductory level and connecting properties with practical implications. This second edition offers enhanced coverage of biological, pharmaceutical, and electrolyte applications including osmotic pressure, solid solubility, and coupled reactions. Throughout the text, topics are organized to implement hierarchical instruction with increasing levels of detail. Content requiring deeper levels of theory is clearly delineated in separate sections and chapters. Less complex empirical model approaches have been moved forward to provide introductory practice with concepts and to provide motivation for understanding models more fully. The approach also provides more instructor flexibility in selecting topics to cover. Learning objectives have been clearly stated for each chapter along with chapter summaries including “important equations” to enhance student focus. Every chapter includes practice problems with complete solutions available online, as well as numerous homework problems. Online supplements include practice tests spanning many years, coursecasts describing difficult concepts or how to use computational tools, ConcepTests to quickly check comprehension, and objective lists that can be customized for greater detail. We also recommend the related resources available at the www.learncheme.com. xvii xviii Preface Unique features of the text include the level of pedagogical development of excess function models and electrolytes. For mixture models, the key assumptions and derivation steps are presented, stimulating readers to consider how the molecular phenomena are represented. For electrolytes and biological systems, the text makes connections between pH and speciation and provides tools for rapidly estimating concentrations of dissociated species. We emphasize speciation and problem solving in this introduction, instead of focusing on advanced theories of electrolyte activity. The material is written at an intermediate level to bridge students from the introductions in chemistry to the more complex models of electrolytes provided by process simulators. We have created a number of homework problems with many variants, intending that different parts can be assigned to different classes or groups, not intending that each student work all parts. NOTES TO STUDENTS Thermodynamics is full of terminology and defined properties. Please note that the textbook provides a glossary and a summary of notation just before Unit I. Also consider the index a resource. We consider the examples to be an integral part of the text, and we use them to illustrate important points. Examples are often cross-referenced and are therefore listed in the table of contents. We enclose important equations in boxes and we use special notation by equation numbers: (*) means that the equation assumes temperature-independent heat capacity; (ig) means the equation is limited to ideal gases. We include margin notes to highlight important concepts or supplemental information. Computer programs facilitate the solutions to homework problems, but they should not be used to replace an understanding of the material. Computers are tools for calculating, not for thinking. To evaluate your understanding, we recommend that you know how to solve the problem by hand calculations. If you do not understand the formulas in the programs it is a good indication that you need to do more studying before using the program so that the structure of the program makes sense. It is also helpful to rework example problems from the text using the software. ACKNOWLEDGMENTS As the above quote from Mark Twain alludes, we are indebted to many others, from informal hallway conversations at meetings, to e-mail messages with suggestions and errata, to classroom testing. In many cases, we are merely the conveyors of others’ suggestions. In particular, for the first edition, Dave Hart, Joan Brennecke, Mike Matthews, Bruce Poling, Ross Taylor, and Mark Thies worked with early versions of the text. We have benefited from classroom testing of the second edition by Margot Vigeant, Victor Vasquez, and Joan Brennecke. We have benefited from reviews by Keith Johnston, Ram Gupta, John O’Connell, Mike Greenfield (electrolytes), Andre Anderko (electrolytes), and Paul Mathias (electrolytes). We have adapted some example problems developed by John O’Connell at the NSF BioEMB Workshop, San Jose, CA, 2010. CTL would like to thank Ryoko Yamasaki for her work in typing many parts of the first edition manuscript and problem solutions. CTL also thanks family members Gail, Nicolas, and Adrienne for their patience, as many family sacrifices helped make this book possible. JRE thanks family members Guliz, Serra, and Eileen for their similar forbearance. We acknowledge Dan Friend and NIST, Boulder, for contributions to the steam tables and thermodynamic charts. Lastly, we acknowledge the influences of the many authors of previous thermodynamics texts. We hope we have done justice to this distinguished tradition, while simultaneously bringing deeper insight to a broader audience. ABOUT THE AUTHORS J. Richard Elliott is Professor of Chemical Engineering at the University of Akron in Ohio. He has taught courses ranging from freshman tools to senior process design as well as thermodynamics at every level. His research interests include: thermodynamics of polymer solutions and hydrogen bonding using molecular simulations and perturbation theory; thermodynamics of supercritical fluids and hydrocarbon processing; biorefining pretreatments; and experimental phase equilibrium measurements. He has worked with the NIST lab in Boulder and ChemStations in Houston. He holds a Ph.D. in chemical engineering from Pennsylvania State University. [email protected] Carl T. Lira is Associate Professor in the Department of Chemical Engineering and Materials Science at Michigan State University. He teaches thermodynamics at all levels, chemical kinetics, and material and energy balances. His research accomplishments include experimental measurements and modeling for liquid metals, supercritical fluids, adsorptive separations, and liquid-vapor, solid-liquid, and liquid-liquid phase equilibria. Currently, Professor Lira specializes in the study of thermodynamic properties of bio-derived fuels and chemicals via experiments and molecular simulations, and he collaborates in the MSU Reactive Distillation Facility. He has been recognized with the Amoco Excellence in Teaching Award, and multiple presentations of the MSU Withrow Teaching Excellence Award. He holds a B.S. from Kansas State University, and an M.S. and Ph.D. from the University of Illinois, Champaign-Urbana, all in chemical engineering. [email protected] xix GLOSSARY Adiabatic—condition of zero heat interaction at system boundaries. Association—description of complex formation where all molecules in the complex are of the same type. Azeotrope—mixture which does not change composition upon vapor-liquid phase change. Barotropy—the state of a fluid in which surfaces of constant density (or temperature) are coincident with surfaces of constant pressure. Binodal—condition of binary phase equilibrium. Dead state—a description of the state of the system when it is in equilibrium with the surroundings, and no work can be obtained by interactions with the surroundings. Diathermal—heat conducting, and without thermal resistance, but impermeable to mass. Efficiency—see isentropic efficiency, thermal efficiency, thermodynamic efficiency. EOS—Equation of state. Fugacity—characterizes the escaping tendency of a component, defined mathematically. Heteroazeotrope—mixture that is not completely miscible in all proportions in the liquid phase and like an azeotrope cannot be separated by simple distillation. The heteroazeotropic vapor condenses to two liquid phases, each with a different composition than the vapor. Upon partial or total vaporization, the original vapor composition is reproduced. Infinite dilution—description of a state where a component’s composition approaches zero. Irreversible—a process which generates entropy. Isenthalpic—condition of constant enthalpy. Isentropic—condition of constant entropy. Isentropic efficiency—ratio characterizing actual work relative to ideal work for an isentropic process with the same inlet (or initial) state and the same outlet (or final) pressure. See also thermodynamic efficiency, thermal efficiency. Isobaric—condition of constant pressure. xxi xxii Glossary Isochore—condition of constant volume. See isosteric. Isopiestic—constant or equal pressure. Isopycnic—condition of equal or constant density. Isolated—A system that has no interactions of any kind with the surroundings (e.g. mass, heat, and work interactions) is said to be isolated. Isosteric—condition of constant density. See isochore. Isothermal—condition of constant temperature. LLE—liquid-liquid equilibria. Master equation—U(V,T). Measurable properties—variables from the set {P, V, T, CP, CV} and derivatives involving only {P, V, T}. Metastable—signifies existence of a state which is non-equilibrium, but not unstable, e.g., superheated vapor, subcooled liquid, which may persist until a disturbance creates movement of the system towards equilibrium. Nozzle—a specially designed device which nearly reversibly converts internal energy to kinetic energy. See throttling. Polytropic exponent—The exponent n in the expression PV n = constant. Quality—the mass fraction of a vapor/liquid mixture that is vapor. rdf—radical distribution function. Reference state—a state for a pure substance at a specified (T,P) and type of phase (S,L,V). The reference state is invariant to the system (P, T ) throughout an entire thermodynamic problem. A problem may have various standard states, but only one reference state. See also standard state. Sensible heat changes—heat effects accompanied by a temperature change. Specific heat—another term for CP or CV with units per mass. Specific property—an intensive property per unit mass. SLE—solid-liquid equilibria. Solvation—description of complex formation where the molecules involved are of a different type. Spinodal—condition of instability, beyond which metastability is impossible. Standard conditions—273.15 K and 0.1 MPa (IUPAC), standard temperature and pressure. Standard state—a state for a pure substance at a specified (T,P) and type of phase (S,L,V). The standard state T is always at the T of interest for a given calculation within a problem. As the T of the system changes, the standard state T changes. The standard state P may be a fixed P or may be the P of the system. Gibbs energies and chemical potentials are commonly calculated relative to the standard state. For reacting systems, enthalpies and Gibbs energies of formation are commonly tabulated at a fixed pressure of 1 bar and 298.15 K. A temperature correction must be applied to calculate the standard state value at the temperature of interest. A problem may have various standard states, but only one reference state. See also reference state. State of aggregation—solid, liquid, or gas. Glossary xxiii Steady-state—open flow system with no accumulation of mass and where state variables do not change with time inside system boundaries. STP—standard temperature and pressure, 273.15 K and 1 atm. Also referred to as standard conditions. Subcooled—description of a state where the temperature is below the saturation temperature for the system pressure, e.g., subcooled vapor is metastable or unstable, subcooled liquid is stable relative to the bubble-point temperature; superheated vapor is stable, superheated liquid is metastable or unstable relative to the dew-point temperature; subcooled liquid is metastable or unstable relative to the fusion temperature. Superheated—description of a state where the temperature is above the saturation temperature for the system pressure. See subcooled. Thermal efficiency—the ratio or work obtained to the heat input to a heat engine. No engine may have a higher thermal efficiency than a Carnot engine. Thermodynamic efficiency—ratio characterizing actual work relative to reversible work obtainable for exactly the same change in state variables for a process. The heat transfer for the reversible process will differ from the actual heat transfer. See also isentropic efficiency, thermal efficiency. Throttling—a pressure drop without significant change in kinetic energy across a valve, orifice, porous plug, or restriction, which is generally irreversible. See nozzle. Unstable—a state that violates thermodynamic stability, and cannot persist. See also metastable, spinodal. VLE—vapor-liquid equilibrium. Wet steam—a mixture of water vapor and liquid. NOTATION General Symbols G Intensive Gibbs energy a g Gravitational constant (9.8066 m/s2) or radial distribution function gc Gravitational conversion factor (1 kg-m/N-s2) (32.174[(1bm-ft)/s2]/1bf) H Intensive enthalpy Ka Reaction equilibrium constant k Boltzmann’s constant R/NA kij Binary interaction coefficient (Eqn. 1.8, 15.9) K Distribution coefficient (vaporliquid, liquid-liquid, etc.) A b Activity, or dimensional equation of state parameter or energetic parameter, or heat capacity or other constant Intensive Helmholtz energy, or dimensionless constant for equation of state, or Antoine, Margules, or other constant Dimensional equation of state parameter or heat capacity or other constant B Virial coefficient, or dimensionless constant for other equation of state, or Antoine or other constant C, c,... Constants, c is a shape factor for the ESD equation of state M Generic property or molarity when used as units CP Intensive constant pressure heat capacity Mw Molecular weight m Mass (energy balances), molality (electrolytes) N Number of molecules n Number of moles NA Avogadro’s number = 6.0221 E 23 mol1 P Pressure CV Intensive constant volume heat capacity F Feed f Pure fluid fugacity fˆi Fugacity of component in mixture xxv xxvi Notation Q Extensive heat transfer across system boundary Parameter for Peng-Robinson equation of state Q Intensive heat transfer across system boundary Universal free volume fraction q Quality (mass% vapor) i Chemical potential for a component in a mixture R Gas constant (8.31446 cm3-MPa/mole-K) Acentric factor Pure fluid fugacity coefficient S Intensive entropy T Temperature ˆ i Component fugacity coefficient in a mixture U Intensive internal energy u Pair potential energy function Osmotic coefficient or electric potential V Intensive volume i Volume fraction of component i v Velocity Intensive density W Extensive work done at boundary Molecular diameter W Intensive work done at boundary Reaction coordinate Z Compressibility factor Operators z Height Denotes change: (1) for a closed system, denotes (final state – initial state); (2) for an open steady-state system, denotes (outlet state – inlet state) ln Natural logarithm (base e) log Common logarithm (base 10) Partial differential Cumulative product operator Cumulative summation Greek Symbols Isobaric coefficient of thermal expansion, also Peng-Robinson equation of state parameter, and also an ESD equation of state variable 1/kT, where k is Boltzmann’s constant and T is temperature. Potential energy parameter Thermal , compressor or pump C, turbine or expander E, efficiency, or reduced density, Pb/V Special Notation ig Equation applies to ideal gas only CP /CV * i Activity coefficient Equation assumes heat capacity is temperature-independent. T Isothermal compressibility S Isentropic compressibility as in H i Partial molar property _ as in U Extensive property ˆ Mixture property Notation Specifies units for variable Equivalence or definition Approximately equal to Subscripts xxvii * Henry’s law (rational) standard state E Excess property for a mixture f Property at final state fus Fusion (melting) process b Property at normal boiling point temperature i Property at initial state EC Expansion/contraction work Denotes phase of interest f Denotes property for formation of molecule from atoms in their naturally occurring pure molecular form ig Ideal gas property in Property at inlet (open system) is Ideal solution property gen Generated entropy L Liquid phase i Component in a mixture out Property at outlet (open system) m Property at melting point S Solid phase mix Used with to denote property change on mixing sat Saturation property V Vapor phase r Property reduced by critical constant vap Vaporization process R Reference state S Shaft work Superscripts Infinite dilution Molality standard state o Standard state (usually pure) CHAPTER BASIC CONCEPTS 1 “Aside from the logical and mathematical sciences, there are three great branches of natural science which stand apart by reason of the variety of far reaching deductions drawn from a small number of primary postulates. They are mechanics, electromagnetics, and thermodynamics. These sciences are monuments to the power of the human mind; and their intensive study is amply repaid by the aesthetic and intellectual satisfaction derived from a recognition of order and simplicity which have been discovered among the most complex of natural phenomena... Yet the greatest development of applied thermodynamics is still to come. It has been predicted that the era into which we are passing will be known as the chemical age; but the fullest employment of chemical science in meeting the various needs of society can be made only through the constant use of the methods of thermodynamics.” Lewis and Randall (1923) Lewis and Randall eloquently summarized the broad significance of thermodynamics as long ago as 1923. They went on to describe a number of the miraculous scientific developments of the time and the relevant roles of thermodynamics. Historically, thermodynamics has guided the development of steam engines, refrigerators, nuclear power plants, and rocket nozzles, to name just a few. The principles remain important today in the refinement of alternative refrigerants, heat pumps, and improved turbines, and also in technological advances including computer chips, superconductors, advanced materials, fermentations, biological cycles, and bioengineered pharmaceuticals. These latter-day “miracles” might appear to have little to do with power generation and refrigeration cycles at first thought. Nevertheless, as Lewis and Randall point out, the implications of the postulates of thermodynamics are far-reaching and will continue to be important in the development of even newer technologies. Much of modern thermodynamics focuses on characterization of the properties of mixtures, as their constituents partition into stable phases or inhomogeneous domains, and react. The capacity of thermodynamics to bring “quantitative precision in place of the old, vague ideas”1 is as germane today as it was then. 1. Lewis, G.N., Randall, M. 1923. Thermodynamics and the Free Energy of Chemical Substances, New York: McGraw-Hill. 3 4 Unit I First and Second Laws Before overwhelming you with the details that comprise thermodynamics, we outline a few “primary postulates” as clearly as possible and put them into the context of what we will refer to as classical equilibrium thermodynamics. In casual terms, our primary premises can be expressed as follows: 1. You can’t get something for nothing. (Energy is conserved.) 2. Maintaining order requires work. (Entropy generation leads to lost work.)2 Occasionally, it may seem that we are discussing principles that are much more sophisticated. But the fact is that all of our discussions can be reduced to these fundamental principles. The first principle is a casual statement of the first law of thermodynamics (conservation of energy) which will be introduced in Chapters 2 and 3. The second principle is a casual statement of the second law of thermodynamics (entropy balance) which will be introduced in Chapter 4. When you find yourself in the midst of a difficult problem, it may be helpful to remember the underlying principles. We will see that coupling these two principles with some slightly sophisticated reasoning (mathematics included) leads to many clear and reliable insights about a wide range of subjects from energy crises to high-tech materials, from environmental remediation to biosynthesis. The bad news is that the level of sophistication required is not likely to be instantly assimilated by the average student. The good news is that many students have passed this way before, and the proper trail is about as well marked as one might hope. There is less-than-universal agreement on what comprises “thermodynamics.” If we simply take the word apart, “thermo” sounds like “thermal,” which ought to have something to do with heat, temperature, or energy. “Dynamics” ought to have something to do with movement. And if we could just leave the identification of thermodynamics as the study of “energy movements,” it would be sufficient for the purposes of this text. Unfortunately, such a definition would not clarify what distinguishes thermodynamics from, say, transport phenomena or kinetics, so we should spend some time clarifying the definition of thermodynamics in this way before moving on to the definitions of temperature, heat, energy, and so on. The definition of thermodynamics as the study of energy movements has evolved considerably to include classical equilibrium thermodynamics, quantum thermodynamics, statistical thermodynamics, and irreversible thermodynamics as well as nonequilibrium thermodynamics. Classical thermodynamics has the general connotation of referring to the implications of constraints related to multivariable calculus as developed by J.W. Gibbs. We spend a significant effort applying these insights in developing generalized equations for the thermodynamic properties of pure substances. Statistical thermodynamics focuses on the idea that knowing the precise states of 1023 atoms is not practical and prescribes ways of computing the average properties of interest based on very limited measurements. We touch on this principle in our introduction to entropy, in our kinetic theory and molecular dynamics, and in the formulation of the internal energy relative to the intermolecular potential energy. We generally refrain from detailed formulation of all the statistical averages, however, maintaining the focus on simple concepts of molecular interactions. Irreversible thermodynamics and nonequilibrium thermodynamics emphasize the ways that local concentrations of atoms and energy evolve over periods of time. At this point, it becomes clear that such a broad characterization of thermodynamics would overlap with transport phenomena and kinetics in a way that would begin to be confusing at the introductory level. Nevertheless, these fields of study represent legitimate subtopics within the general realm of thermodynamics. 2. The term “lost work” refers to the loss of capability to perform useful work, and is discussed in more detail in Sections 2.4 on page 42, 4.2 on page 132, and 4.3 on page 142. Section 1.1 Introduction 1.1 5 INTRODUCTION These considerations should give you some idea of the potential range of applications possible within the general study of thermodynamics. This text will try to find a happy medium. One general unifying principle about the perspective offered by thermodynamics is that there are certain properties that are invariant with respect to time. For example, the process of diffusion may indicate some changes in the system with time, but the diffusion coefficient is a property which only depends on a temperature, density, and composition profile. A thermodynamicist would consider the diffusion process as something straightforward given the diffusion coefficient, and focus on understanding the diffusion coefficient. A transport specialist would just estimate the diffusion coefficient as best as he could and get on with it. A kineticist would want to know how fast the diffusion was relative to other processes involved. In more down-to-earth terms, if we were touring about the countryside, the thermodynamicists would want to know where we were going, the transport specialists would want to know how long it takes to get there, and the kineticists would want to know how fast the fuel was running out. In thermodynamics we utilize a few basic concepts: energy, entropy, and equilibrium. The ways in which these are related to one another and to temperature, pressure, and density are best understood in terms of the connections provided by molecular mechanisms. These connections, in turn, can be summarized by the thermodynamic model (e.g., ideal gas), our quantitative description of the substance. Showing how energy and entropy couple with molecular characteristics to impact chemical process applications is the primary goal of this text. These insights should stick with you long after you have forgotten how to estimate any particular thermodynamic property, a heat capacity or activity coefficient, for example. We will see how assuming a thermodynamic model and applying the rules of thermodynamics leads to precise and extremely general insights relevant to many applications. A general theme throughout the text (and arguably throughout engineering) is: observe, predict, test, and evaluate. The prediction phase usually involves a model equation. Testing and evaluation expose limitations of the prospective model, which leads to a new cycle of observation, prediction... We terminate this hierarchy at an introductory level, but it never really ends. Extending this hierarchy is the source of innovation that must serve you for the next 50 years. Chapter Objectives: You Should Be Able to... 1. Explain the definitions and relations between temperature, molecular kinetic energy, molecular potential energy and macroscopic internal energy, including the role of intermolecular potential energy and how it is modeled. Explain why the ideal gas internal energy depends only on temperature. 2. Explain the molecular origin of pressure. 3. Apply the vocabulary of thermodynamics with words such as the following: work, quality, interpolation, sink/reservoir, absolute temperature, open/closed system, intensive/extensive property, subcooled, saturated, superheated. 4. Explain the advantages and limitations of the ideal gas model. 5. Sketch and interpret paths on a P-V diagram. 6. Perform steam table computations like quality determination, double interpolation. 6 Unit I First and Second Laws 1.2 THE MOLECULAR NATURE OF ENERGY, TEMPERATURE, AND PRESSURE Energy is a term that applies to many aspects of a system. Its formal definition is in terms of the capability to perform work. We will not quantify the potential for work until the next chapter, but you should have some concept of work from your course in introductory physics. Energy may take the form of kinetic energy or potential energy, and it may refer to energy of a macroscopic or a molecular scale. Energy is the sum total of all capacity for doing work that is associated with matter: kinetic, potential, submolecular (i.e., molecular rearrangements by reaction), or subatomic (e.g., ionization, fission). Kinetic energy is the energy associated with motion of a system. Motion can be classified as translational, rotational, or vibrational. Temperature is related to the “hotness” of a substance, but is fundamentally related to the kinetic energy of the constituitive atoms. Potential energy is the energy associated with a system due to its position in a force field. ! Energy will be tabulated relative to a convenient reference state. In the study of “energy movements,” we will continually ask, “How much energy is here now, and how much is there?” In the process, we need to establish a point for beginning our calculations. According to the definition above, we might intuitively represent zero internal energy by a perfect vacuum. But then, knowing the internal energy of a single proton inside the vacuum would require knowing how much energy it takes to make a proton from nothing. Since this is not entirely practical, this intuitive choice is not a good engineering choice usually. This is essentially the line of reasoning that gives rise to the convention of calculating energy changes relative to a reference state. Thus, there is no absolute reference point that is always the most convenient; there are only changes in energy from one state to another. We select reference conditions that are relevant throughout any particular process of interest. Depending on the complexity of the calculation, reference conditions may vary from, say, defining the enthalpy (to be defined later) of liquid water to be zero at 0.01C (as in the steam tables) to setting it equal to zero for the molecular hydrogen and oxygen at 1 bar and 298.15 K (as in the heat of reaction), depending on the situation. Since this text focuses on changes in kinetic energy, potential energy, and energies of reaction, we need not specify reference states any more fundamental than the elements, and thus we do not consider subatomic particles. Kinetic Energy and Temperature Kinetic energy is commonly introduced in detail during introductory physics as ½ mv2, where m is the mass of the object and v is the object velocity. Atomic species that make up solids are frozen in localized positions, but they are continuously vibrating with kinetic energy. Fluids such as liquids and gases are not frozen into fixed positions and move through space with kinetic energy and collide with one another. ! Temperature primarily reflects the kinetic energy of the molecules. The most reliable definition of temperature is that it is a numerical scale for uniquely ordering the “hotness” of a series of objects.3 However, this “hotness” is coupled to the molecular kinetic energy of the constituent molecules in a fundamental way. The relation between kinetic energy and temperature is surprisingly direct. When we touch a hot object, the kinetic energy of the object is transferred to our hand via the atoms vibrating at the surface. Temperature is proportional to the 3. Denbigh, K., 1971. The Principles of Chemical Equilibrium, London: Cambridge University Press, p. 9. Section 1.2 The Molecular Nature of Energy, Temperature, and Pressure 7 average molecular kinetic energy. The expression is easiest to use in engineering on a molar basis. For a monatomic substance Mw 2 T = -------- v (for 3D) 3R Mw 2 T 2D = -------- v (for 2D) monatomic fluid. 2R 1.1 where brackets denote an average, and Mw is the molecular weight. We use a subscript for the temperature of 2D motion to avoid confusion with the more common 3D motion. The differences between 2D and 3D temperature are explained on page 22. For a polyatomic molecule, the temperature is coupled to the average velocity of the individual atoms, but some of the motion of the bonded atoms results in vibrations and rotations rather than a direct translation of the center of mass and thus it is not directly related to the velocity of the center of mass. (See Section 7.10 on page 276.) ! Check your units when using this equation. Eqn. 1.1 is applicable to any classical monatomic system, including liquids and solids. This means that for a pure system of a monatomic ideal gas in thermal equilibrium with a liquid, the average velocities of the molecules are independent of the phase in which they reside. We can infer this behavior by envisioning gas atoms exchanging energy with the solid container walls and then the solid exchanging energy with the liquid. At equilibrium, all exchanges of energy must reach the same kinetic energy distribution. The liquid molecular environment is different from the gas molecular environment because liquid molecules are confined to move primarily within a much more crowded environment where the potential energies are more significant. When a molecule’s kinetic energy is insufficient to escape the potential energy (we discuss the potential energy next) due to molecular attractiveness, the atoms simply collide with a higher frequency in their local environment. What happens if the temperature is raised such that the liquid molecules can escape the potential energies of the neighbors? We call this phenomenon “boiling.” Now you can begin to understand what temperature is and how it relates to other important thermodynamic properties. We are guaranteed that a universal scale of temperature can be developed because of the zeroth law of thermodynamics: If two objects are in equilibrium with a third, then they are in equilibrium with one another as we discussed in the previous paragraph. The zeroth law is a law in the sense that it is a fact of experience that must be regarded as an empirical fact of nature. The significance of the zeroth law is that we can calibrate the temperature of any new object by equilibrating it with objects of known temperature. Temperature is therefore an empirical scale that requires calibration according to specific standards. The Celsius and Fahrenheit scales are in everyday use. The conversions are: 5 T in C = --- T in F – 32 9 When we perform thermodynamic calculations, we must usually use absolute temperature in Kelvin or Rankine. These scales are related by T in K = T in C + 273.15 T in °R = T in F + 459.67 (T in R) 1.8 · (T in K) The absolute temperature scale has the advantage that the temperature can never be less than absolute zero. This observation is easily understood from the kinetic perspective. The kinetic ! Thermodynamic calculations use absolute temperature in °R or K. 8 Unit I First and Second Laws energy cannot be less than zero; if the atoms are moving, their kinetic energy must be greater than zero. Potential Energy Solids and liquids exist due to the intermolecular potential energy (molecular “stickiness’) of atoms. If molecules were not “sticky” all matter would be gases or solids. Thus, the principles of molecular potential energy are important for developing a molecular perspective on the nature of liquids, solids, and non-ideal gases. Potential energy is associated with the “work” of moving a system some distance through a force field. On the macroscopic scale, we are well aware of the effect of gravity. As an example, the Earth and the moon are two spherical bodies which are attracted by a force which varies as r2. The potential energy represents the work of moving the two bodies closer together or farther apart, which is simply the integral of the force over distance. (The force is the negative derivative of potential with respect to distance.) Thus, the potential function varies as r1. Potential energies are similar at the microscopic level except that the forces vary with position according to different laws. The gravitational attraction between two individual atoms is insignificant because the masses are so small. Rather, the important forces are due to the nature of the atomic orbitals. For a rigorous description, the origin of the intermolecular potential is traced back to the solution of Schrödinger’s quantum mechanics for the motions of electrons around nuclei. However, we do not need to perform quantum mechanics to understand the principles. Intermolecular Potential Energy ! Engineering model potentials permit representation of attractive and repulsive forces in a tractable form. Atoms are composed of dense nuclei of positive charge with electron densities of negative charge built around the nucleus in shells. The outermost shell is referred to as the valence shell. Electron density often tends to concentrate in lobes in the valence orbitals for common elements like C, N, O, F, S, and Cl. These lobes may be occupied by bonded atoms that are coordinated in specific geometries, such as the tetrahedron in CH4, or they may be occupied by unbonded electron pairs that fill out the valence as in NH3 or H2O, or they may be widely “shared” as in a resonance or aromatic structure. These elements (H, C, N, O, F, S, Cl) and some noble gases like He, Ne, and Ar provide virtually all of the building blocks for the molecules to be considered in this text. By considering the implications of atomic structure and atomic collisions, it is possible to develop the following subclassifications of intermolecular forces: 1. Electrostatic forces between charged particles (ions) and between permanent dipoles, quadrupoles, and higher multipoles. 2. Induction forces between a permanent dipole (or quadrupole) and an induced dipole. 3. Forces of attraction (dispersion forces) due to the polarizability of electron clouds and repulsion due to prohibited overlap. 4. Specific (chemical) forces leading to association and complex formation, especially evident in the case of hydrogen bonding. ! Attractive forces and potential energies are negative. Repulsive forces and potential energies are positive. Attractive forces are quantified by negative numerical values, and repulsive forces will be characterized by positive numerical values. To a first approximation, these forces can be characterized by a spherically averaged model of the intermolecular potential (aka. “potential” model). The potential, u(r), is the work (integral of force over distance) of bringing two molecules from infinite distance to a specific distance, r. When atoms are far apart (as in a low-pressure gas), they do not sense one another and interaction energy approaches zero. When the atoms are within about two diameters, they attract, resulting in a negative energy of interaction. Because they have finite Section 1.2 The Molecular Nature of Energy, Temperature, and Pressure 9 size, as they are brought closer, they resist overlap. Thus, at very close distances, the forces are repulsive and create very large positive potential energies. These intuitive features are illustrated graphically in Fig. 1.1. The discussion below provides a brief background on why these forces exist and how they vary with distance. Electrostatic Forces The force between two point charges described by Coulomb’s Law is very similar to the law of gravitation and should be familiar from elementary courses in chemistry and physics, qi q j F --------r2 where qi and qj are the charges, and r is the separation of centers. Upon integration, u = potential energy is proportional to inverse distance, qi qj u --------r F dr , the 1.2 If all molecules were perfectly spherical and rigid, the only way that these electrostatic interactions could come into play is through the presence of ions. But a molecule like NH3 is not perfectly spherical. NH3 has three protons on one side and a lobe of electron density in the unbonded valence shell electron pair. This permanent asymmetric distribution of charge density is modeled mathematically with a dipole (+ and – charge separation) on the NH3 molecule.4 This means that ammonia molecules lined up with the electrons facing one another repel while molecules lined up with the electrons facing the protons will attract. Since electrostatic energy drops off as r1, one might expect that the impact of these forces would be long-range. Fortunately, with the close proximity of the positive charge to the negative charge in a molecule like NH3, the charges tend to cancel one another as the molecule spins and tumbles about through a fluid. This spinning and tumbling makes it reasonable to consider a spherical average of the intermolecular energy as a function of distance that may be constructed by averaging over all orientations between the molecules at each distance. In a large collection of molecules randomly distributed relative to one another, this averaging approach gives rise to many cancellations, and the net impact is approximately – dipole u dipole – dipole ------------------r 6 kT 1.3 where k = R/NA is Boltzmann’s constant, related to the gas constant, R, and Avogadro’s number, NA. This surprisingly simple result is responsible for a large part of the attractive energy between polar molecules. This energy is attractive because the molecules tend to spend somewhat more time lined up attractively than repulsively, and the r6 power arises from the averaging that occurs as the molecules tumble and the attractive forces decrease with separation. A key feature of dipole-dipole forces is the temperature dependence. 4. The dipole is a model of the charge distribution on the molecule, and it is thus an approximate description of the charge distribution. 10 Unit I First and Second Laws Induction Forces When a molecule with a permanent dipole approaches a molecule with no dipole, the positive charge of the dipolar molecule will tend to pull electron density away from the nonpolar molecule and “induce” a dipole moment into the nonpolar molecule. The magnitude of this effect depends on the strength of the dipole and how tightly the orbitals of the nonpolar molecule constrain the electrons spatially in an electric field, characterized by the “polarizability.”5 For example, the pi bonding in benzene makes it fairly polarizable. A similar consideration of the spherical averaging described in relation to electrostatic forces results again in a dependence of r6 as approximately – in u in ---------r6 1.4 Disperse Attraction Forces (Dispersion Forces) ! The r 6 dependence of attractive forces has a theoretical basis. When two nonpolar molecules approach, they may also induce dipoles into one another owing to fluctuating distributions of electrons. Their dependence on radial distance may be analyzed and gives the form for the attractive forces: – att att ----------u disp r6 1.5 Note that dipole-dipole, induction, and dispersion forces all vary as r–6. Repulsive Forces The forces become repulsive rapidly as radial distance decreases, and quickly outweighs the attractive force as the atoms are forced into the same space. A common empirical equation is rep rep ---------u disp r 12 1.6 Engineering Potential Models Based on the forms of these electrostatic, induction, and dispersion forces, it should be easy to appreciate the form of the Lennard-Jones potential in Fig. 1.1. Other approximate models of the potential function are possible, such as the square-well potential or the Sutherland potential also shown in Fig. 1.1. These simplified potential models are accurate enough for many applications. The key features of all of these potential models are the representation of the size of the molecule by the parameter and the attractive strength (i.e. “stickiness”) by the parameter . We can gain considerable insight about the thermodynamics of fluids by intuitively reasoning about the relatively simple effects of size and stickiness. For example, if we represent molecules by lumping together all the atomic sites, a large molecule like buckminsterfullerene (solid at room temperature) would have a larger value for and than would methane (gas at room temperature). Water and methane are about the same size, but their difference in boiling temperature indicates a large difference in their stickiness. Considering the molecular perspective, it should become apparent that water 5. The polarizability is the linear proportionality constant in a model of how easily a dipole is “induced” when the molecule is placed in an electric field. Section 1.2 The Molecular Nature of Energy, Temperature, and Pressure 1.5 1 0.5 11 The Lennard-Jones potential. u(r) 4[(/r)12 – (/r)6] u/ 0 Note that >0 is the depth of the well, and is the distance where u(r) = 0. -0.5 -1 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r/ 1.5 1.0 0.5 u/ 0.0 -0.5 -1.0 -1.5 The square-well potential for 1.5. if r u r = – if r 0 if r 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r/ 1.5 1 The Sutherland potential. 0.5 u/ if r ur = – r 6 if r 0 -0.5 -1 -1.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r/ Figure 1.1 Schematics of three engineering models for pair potentials on a dimensionless basis. has a higher boiling temperature because it sticks to itself more strongly than does methane. With these simple insights, you should be able to understand the molecular basis for many macroscopic phenomena. Example 1.1 illustrates several computations for intermolecular potential energy. 12 Unit I First and Second Laws Example 1.1 The energy derived from intermolecular potentials We can develop further appreciation for internal energy by computing the intermolecular potential energy for a well-defined system of molecules. Assume the Lennard-Jones potential model with = 0.36 nm and = 1.38E-21 J: (a) Compute the potential energy for two molecules located at positions (0,0) and (0, 0.4 nm). (b) Suppose a third molecule was located at (0.5,0). Compute the potential energy for the system. (c) To develop a very crude insight on the methods of averaging, we can think of the average potential energy as defining an average distance between the molecules. As the volume expands, the average distance between molecules increases and the magnitude of the average potential energy decreases in accordance with the Lennard-Jones model. For the potential energy from (b), compute the average distance, <r>, that corresponds to the average potential energy for this system of molecules. (d) Suppose the volume of the system in (c) expands by a factor of two. How would that affect the average distance, <r>, and what would you estimate as the new intermolecular energy? (e) Assume approximately four molecules can fit around a central molecule in a liquid before it is too crowded and another layer starts to build up. Assuming the Lennard-Jones energy is practically zero beyond the first layer (i.e. ignore all but the first layer), and the average distance between the central molecule and its four neighbors is <r> = 0.55 nm, estimate the intermolecular energy around one single molecule and that for one mole of similar molecules. Solution: (a) The distance is r12 = [ (0 – 0)2 + (0.4 – 0)2 ]½ = 0.4 nm. u(r12) = 4(1.38E-21)[(0.36/0.40)12 – (0.36/0.40)6] = –1.375E-21 (b) The distance r13 = [ (0.5 – 0)2 + (0.5 – 0)2 ]½ = 0.5. So u(r13) = –0.662E-21. But wait, there’s more! r23 = [ (0.5 – 0)2 + (0 – 0.4)2 ]½ = 0.6403. So, u(r23) = –0.169E-21. The total intermolecular energy is: –(1.375+0.662+0.169)(1E-21) = –2.206E-21J. (c) The average intermolecular energy for these three pairs is: <u> = –2.206E-21/3 = –0.735E-21J. Matching this value of <u> by using a solver to adjust <r> in the Lennard-Jones model gives <r> = 0.4895 nm. (d) Volume is related to the cubic of length. Expanding the volume by a factor of 2 changes the r-coordinates by a factor of 21/3. So, <r> = 0.4895(21/3) = 0.6167, and <u> = –0.210E-21J. (e) For <r> = 0.55, <u(r)> = -0.400E-21J and u1 = 4(–0.400E-21) = –1.600E-21 per atom. For Avogadro’s number of such molecules, the summed intermolecular energy becomes uNA = NA<u> = (602.22E21)(–1.600E-21) = –963 J/mole. ! Configurational energy is the potential energy of a system of molecules in their “configuration.” When we sum the potential energy for a collection of molecules, we often call the sum configurational energy to differentiate quantity from the potential energy which is commonly used when discussing atoms or sites. Note that we would need a more complicated potential model to represent the shape of the molecule. Typically, molecules of different shapes are represented by binding together several Section 1.2 The Molecular Nature of Energy, Temperature, and Pressure 13 potentials like those above with each potential site representing one molecular segment. For example, n-butane could be represented by four Lennard-Jones sites that have their relative centers located at distances corresponding to the bond-lengths in n-butane. The potential between two butane molecules would then be the sum of the potentials between each of the individual LennardJones sites on the different molecules. In similar fashion, potential models for very complex molecules can be constructed. Potentials in Mixtures Our discussion of intermolecular potentials has focused on describing single molecules, but it is actually more interesting to contemplate the potential models for different molecules that are mixed together. Note that the square-well model provides a simple way for use to consider only the potential energy of the closest neighbors. We can use the square-well potential as the basis for this analysis and focus simply on the size (ij) and stickiness (ij) of each potential model, where the subscript ij indicates an interaction of molecule i with molecule j. Commonly, we assume that = 1.5 in discussions of the square-well potential, unless otherwise specified. For example, 11 would be the stickiness of molecule 1 to itself, and 12 would be its stickiness to a molecule of type 2 and 21=12. We often calculate the interactions of dissimilar molecules by using combining rules that relate the interaction to the parameters of the sites. Commonly we use combining rules developed by Lorentz and Bertholet. The size parameter for interaction between different molecules is reasonably well represented by 12 (11 22)/ 2 1.7 This rule simply states that the distance between two touching molecules is equal to the radius of the first one plus the radius of the second one. The estimation of the stickiness parameter for interaction between different molecules requires more empirical reasoning. It is conventional to estimate the stickiness by a geometric mean, but to permit flexibility to adjust the approximate rule by adding an adjustable constant that can be refined using experimental measurements, or sometimes using theories like quantum mechanical simulation. For historical reasons, this constant is typically referred to as “kij” or the binary interaction parameter, and defined through the following rule: 12 (11 22)1/2(1 – k12) 1.8 The default value is k12 = 0. Specific (Chemical) Forces Like Hydrogen Bonding What happens when the strength of interaction between two molecules is so strong at certain orientations that it does not make sense to spherically average over it? Hydrogen bonding is an example of such an interaction, as you probably know from an introductory chemistry or biology course. For instance, it would not make sense to spherically average when two atoms preferentially interact in a specific orientation. But, if they were covalently bonded, we would call that a chemical reaction and handle it in a different way. An interesting problem arises when the strength of interaction is too strong to be treated entirely by spherically averaging and too weak to be treated as a normal chemical reaction which forms permanent stable chemical species. Clearly, this problem is difficult and it would be tempting to try to ignore it. In fact, most of this course will deal with theories that treat only what can be derived by spherically averaging. But it should be kept in mind that these specific forces are responsible for some rather important properties, especially in the form of hydrogen bonding. Since many systems are aqueous or contain amides or alcohols, ignoring hydro- 14 Unit I First and Second Laws gen bonding entirely would substantially undermine the accuracy of our conceptual foundation. Furthermore, the concept of favorable energetic interactions between acids and bases can lend broad insights into the mysteries of chemical formulations. As an engineering approach, we can make large adjustments to the spherical nature of these forces such that we can often approximate them with a single characteristic constant to obtain a workable engineering model. Example 1.2 illustrates the concept of combining rules that pervades the entirety of mixture thermodynamics. Example 1.2 Intermolecular potentials for mixtures Methane (CH4) has fewer atoms than benzene (C6H6), so it is smaller. Roughly, the diameter of methane is 0.36 nm and that of benzene is 0.52 nm. Similarly, methane’s boiling temperature is lower so its stickiness must be smaller in magnitude. A crude approximation is /k = Tc/1.25, where Tc is the critical temperature listed on the back flap. Not knowing what to assume for k12, we may consider three possibilities: k12 = 0, k12 < 0, k12 > 0. To illustrate, sketch on the same pair of axes the potential models for methane and benzene, assuming that the k12 parameter is given by: (a) k12 = 0 (b) k12 = –0.2 (c) k12 = +0.2. In each case, describe in words what is represented by each numerical value (e.g., favorable interactions, or unfavorable interactions...) Assume the square-well potential with = 1.5. Solution: Following the suggested estimation formula, /k = 190.6/1.25 = 152.5 for methane and 562.2/1.25 = 449.8 for benzene. Applying Eqns. 1.7 and Eqn. 1.8, we obtain Fig. 1.2 where k12 = 0.0 refers to case (a); –0.2 refers to the well location for case (b); and +0.2 refers to case (c). 200 100 u/k (K) 0 -100 u 11 -200 k 12 = +0.2 k 12 = 0 u 12 -300 -400 k 12 = -0.2 u 22 -500 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r (nm) Figure 1.2 Sketch of intermolecular square-well potential models for a mixture of methane and benzene for = 1.5 as explained in Example 1.1. Description of the interactions: Case (a) corresponds to the molecular interactions being relatively neutral towards one another. This is the default assumption (i.e. k12 = 0). This would be the best description of methane + benzene (among these three choices) because both are nonpolar. Case (b) corresponds to extremely favorable interactions, as indicated by the deep attractive well and strongly exothermic interaction. An Lewis acid might interact with a Lewis base in this way. Case (c) corresponds to unfavorable interactions. The interactions are not zero exactly, so the molecules do attract one another, but the well for the 1-2 interactions is not as deep as expected from the 1-1 and 1-2 interactions. The molecules greatly prefer their own company. A mixture with this type of interaction may lead to liquid-liquid splitting, like oil and water. Section 1.4 Basic Concepts 15 Internal Energy We have taken considerable time to develop the molecular aspects of kinetic and potential energy. These molecular properties are of great importance, but for large-scale macroscopic process system calculations, these microscopic energies are combined and we consider kinetic and configurational energy collectively as the internal energy of the system, which is given the symbol U. It may be somewhat confusing that kinetic and potential energy exist on the macroscopic level and the microscopic level. The potential energy (gravitational potential energy) and kinetic energy of the center of mass of the system are accounted for separately. The internal energy is a function of the temperature and density (the latter for non-ideal gases) of the system, and it does not usually change if the entire system is placed on, say, an airplane where the kinetic and potential energy of the center of mass differ considerably from a stationary position on the ground. This is the convention followed throughout the remainder of Unit I. In Units II and III, we reexamine the molecular potentials as to how they affect the bulk fluid properties. Thus, throughout the remainder of Unit I, when we refer to kinetic and potential energy of a body of fluid as a system, we are referring to the kinetic energy of the center of mass of the system and the gravitational potential energy of the center of mass of the system. 1.3 THE MOLECULAR NATURE OF ENTROPY To be fair to both of the central concepts of thermodynamics, we must mention entropy at this point, in parallel with the mention of energy. Unfortunately, there is no simple analogy that can be drawn for entropy like that for potential energy using the gravitational forces between the Earth and moon. The study of entropy is fairly specific to the study of thermodynamics. What we can say at this point is that entropy has been conceived to account for losses in the prospect of performing useful work. The energy can take on many forms and be completely accounted for without contemplating how much energy has been “wasted” by converting work into something like warm water. To picture how much work this would take, imagine yourself stirring water in a bath tub until the temperature rises by 5°C. Entropy accounts for this kind of wastefulness. It turns out that the generation of such waste can be directly related to the degree of disorder involved in conducting our process. Thus, generation of disorder results in lost work. Furthermore, work that is lost by not maintaining order cannot be converted into useful work. To see the difference between energy and entropy, consider the following example. Oxygen and nitrogen are mixed as ideal gases at room temperature and pressure. How much energy is involved in the mixing process? How much (energy) work must be exerted to separate them again? To completely answer the first question we must understand the ideal gas mixture more completely (Chapter 4). We note that ideal gases are point masses with no potential energy to affect the mixing. Thus, the answer to the first question is that no energy is involved. For the answer to the second question regarding work to separate them, however, we must acknowledge that a considerable effort would be involved. The minimum amount required can be calculated as we will show in Chapter 4. To avoid introducing too many concepts at one time, we defer the quantitative development of the topic until Chapter 4. 1.4 BASIC CONCEPTS The System A system is that portion of the universe which we have chosen to study. ! The sum of microscopic random kinetic energy and intermolecular potential energies comprises the internal energy. 16 Unit I First and Second Laws A closed system is one in which no mass crosses the system boundaries. An open system is one in which mass crosses the system boundaries. The system may gain or lose mass or simply have some mass pass through it. An isolated system is one devoid of interactions of any kind with the surroundings (including mass exchange, heat, and work interactions). ! The placement of system boundaries is a key step in problem solving. System boundaries are established at the beginning of a problem, and simplification of balance equations depends on whether the system is open or closed. Therefore, the system boundaries should be clearly identified. If the system boundaries are changed, the simplification of the mass and energy balance equations should be performed again, because different balance terms are likely to be necessary. These guidelines become more apparent in Chapter 2. In many textbooks, especially those dealing with fluid mechanics, the system is called the control volume. The two terms are synonymous. Equilibrium A system is in equilibrium when there is no driving force for a change of intensive variables within the system. The system is “relaxed” relative to all forces and potentials.6 An isolated system moves spontaneously to an equilibrium state. In the equilibrium state there are no longer any driving forces for spontaneous change of intensive variables. The Mass Balance Presumably, students in this course are familiar with mass balances from an introductory course in material and energy balances. The relevant relation is simply: rate of mass ! The mass balance. accumulation within = system boundaries m· = rate of mass flow into system m· in – inlets – rate of mass flow out of system 1.9 m· out outlets dm where m· = ------- . m· in and m· out are the absolute values of mass flow rates entering and leaving, dt respectively. We may also write dm = dm inlets in – dm out 1.10 outlets 6. We qualify this criterion for the purposes of chemical engineering that there is no driving force for “meaningful” change because most of our systems are technically metastable (in a state of local equilibrium). For example, considering air expansion in a piston/cylinder expansion, we neglect the potential corrosion of the piston/cylinder by air oxidation when we state the system has reached mechanical equilibrium. Section 1.4 Basic Concepts 17 where mass differentials dmin and dmout are always positive. When all the flows of mass are analyzed in detail for several subsystems coupled together, this simple equation may not seem to fully portray the complexity of the application. The general study of thermodynamics is similar in that regard. A number of simple relations like this one are coupled together in a way that requires some training to understand. In the absence of chemical reactions, we may also write a mole balance by replacing mass with moles in the balance. Heat – Sinks and Reservoirs Heat is energy in transit between the source from which the energy is coming and a destination toward which the energy is going. When developing thermodynamic concepts, we frequently assume that our system transfers heat to/from a reservoir or sink. A heat reservoir is an infinitely large source or destination of heat transfer. The reservoir is assumed to be so large that the heat transfer does not affect the temperature of the reservoir. A sink is a special name sometimes used for a reservoir which can accept heat without a change in temperature. The assumption of constant temperature makes it easier to concentrate on the thermodynamic behavior of the system while making a reasonable assumption about the part of the universe assigned to be the reservoir. ! A reservoir is an infinitely large source or destination for heat transfer. The mechanics of heat transfer are also easy to picture conceptually from the molecular kinetics perspective. In heat conduction, faster-moving molecules collide with slower ones, exchanging kinetic energy and equilibrating the temperatures. In this manner, we can imagine heat being transferred from the hot surface to the center of a pizza in an oven until the center of the pizza is cooked. In heat convection, packets of hot mass are circulated and mixed, accelerating the equilibration process. Heat convection is important in getting the heat from the oven flame to the surface of the pizza. Heat radiation, the remaining mode of heat transfer, occurs by an entirely different mechanism having to do with waves of electromagnetic energy emitted from a hot body that are absorbed by a cooler body. Radiative heat transfer is typically discussed in detail during courses devoted to heat transfer. Work Work is a familiar term from physics. We know that work is a force acting through a distance. There are several ways forces may interact with the system which all fit under this category, including pumps, turbines, agitators, and pistons/cylinders. We will discuss the details of how we calculate work and determine its impact on the system in the next chapter. Density Density is a measure of the quantity per unit volume and may be expressed on a molar basis (molar density) or a mass basis (mass density). In some situations, it is expressed as the number of particles per unit volume (number density). Intensive Properties Intensive properties are those properties which are independent of the size of the system. For example, in a system at equilibrium without internal rigid/insulating walls, the temperature and pressure are uniform throughout the system and are therefore intensive properties. Specific properties are the total property divided by the mass and are intensive. For example, the molar volume ([] length3/mole), mass density ([] mass/length3), and specific internal energy ([] energy/mass) are intensive properties. In this text, intensive properties are not underlined. ! The distinction between intensive and extensive properties is key in selecting and using variables for problem solving. 18 Unit I First and Second Laws Extensive Properties Extensive properties depend on the size of the system, for example the volume ([] length3) and energy ([] energy). Extensive properties are underlined; for example, U nU, where n is the number of moles and U is molar internal energy. States and State Properties – The Phase Rule ! The Gibbs phase rule provides the number of state variables (intensive properties) to specify the state of the system. Two state variables are necessary to specify the state of a single-phase pure fluid, that is, two from the set P, V, T, U. Other state variables to be defined later in the text which also fit in this category are molar enthalpy, molar entropy, molar Helmholtz energy, and molar Gibbs energy. State variables must be intensive properties. As an example, specifying P and T permits you to find the specific internal energy and specific volume of steam. Note, however, that you need to specify only one variable, the temperature or the pressure, if you want to find the properties of saturated vapor or liquid. This reduction in the needed specifications is referred to as a reduction in the “degrees of freedom.” As another example in a ternary, two-phase system, the temperature and the mole fractions of two of the components of the lower phase are state variables (the third component is implicit in summing the mole fractions to unity), but the total number of moles of a certain component is not a state variable because it is extensive. In this example, the pressure and mole fractions of the upper phase may be calculated once the temperature and lower-phase mole fractions have been specified. The number of state variables needed to completely specify the state of a system is given by the Gibbs phase rule for a non-reactive system,7: F C P 2 1.11 where F is the number of state variables that can be varied while P phases exist in a system where C is the number of components (F is also known as the number of degrees of freedom). More details on the Gibbs phase rule are given in Chapter 16. Steady-State Open Systems ! Steady-state flow is very common in the process industry. The term steady state is used to refer to open systems in which the inlet and outlet mass flow rates are invariant with time and there is no mass accumulation. In addition, steady state requires that state variables at all locations are invariant with respect to time. Note that state variables may vary with position. Steady state does not require the system to be at equilibrium. For example, in a heat exchanger operating at steady state with a hot and cold stream, each stream has a temperature gradient along its length, and there is always a driving force for heat transfer from the hotter stream to the colder stream. Section 2.13 describes this process in more detail. The Ideal Gas Law ! The ideal gas law is a model that is not always valid, but gives an initial guess. ! An equation of state relates the P-V-T properties of a fluid. The ideal gas is a “model” fluid where the molecules have no attractive potential energy and no size (and thus, no repulsive potential energy). Properties of the ideal gas are calculated from the ideal gas model: PV = nRT (ig) 1.12 Note that scientists who first developed this formula empirically termed it a “law” and the name has persisted, but it should be more appropriately considered a “model.” In the terminology we develop, it is also an equation of state, relating the P-V-T properties of the ideal gas law to one 7. For a reactive system, C is replaced with the number of distinct species minus the number of independent reactions. 19 Section 1.4 Basic Concepts 50 40 o Temperature C 100 200 P (MPa) 30 300 400 20 500 10 0 1 1.5 2 2.5 3 3.5 4 4.5 5 log10 [V (cm 3/mol)] Figure 1.3 Ideal gas behavior at five temperatures. another as shown in Eqn. 1.12 and Fig. 1.3. We know that real molecules have potential energy of attraction and repulsion. Due to the lack of repulsive forces, ideal gas particles can “pass through” one another. Ideal gas molecules are sometimes called “point masses” to communicate this behavior. While the assumptions may seem extreme, we know experimentally that the ideal gas model represents many compounds, such as air, nitrogen, oxygen, and even water vapor at temperatures and pressures near ambient conditions. Use of this model simplifies calculations while the concepts of the energy and entropy balances are developed throughout Unit I. This does not imply that the ideal gas model is applicable to all vapors at all conditions, even for air, oxygen, and nitrogen. Analysis using more complex fluid models is delayed until Unit II. We rely on thermodynamic charts and tables until Unit II to obtain properties for gases that may not be considered ideal gases. Because kinetic energy is the only form of energy for an ideal gas, the internal energy of a monatomic ideal gas is given by summing the kinetic energy of the atoms and then relating this to temperature (c.f. Eqn. 1.1): nN A m v 2 nM w v 2 3 3 Nm v 2 U ig = -------------------- = ------------------------- = ---------------------- = --- nRT = --- NkT (monatomic ideal gas) (ig) 1.13 2 2 2 2 2 The proportionality constant between temperature and internal energy is known as the ideal gas heat capacity at constant volume, denoted CV. Eqn 1.13 shows that CV 1.5R for a monatomic ideal gas. If you refer to the tables of constant pressure heat capacities (CP) on the back flap of the 20 Unit I First and Second Laws text and note that CP CV R for an ideal gas, you may be surprised by how accurate this ultrasimplified theory actually is for species like helium, neon, and argon at 298 K.8 While the equality in Eqn. 1.13 is valid for monatomic fluids only, the functionality U ig U (T ) is universal for all ideal gases. For more multi-atom molecules, the heat capacity depends on temperature because vibrations hold some energy in a manner that depends on temperature. However, the observation that U ig U ig(T ) is true for any ideal gas, not only for ultrasimplified, monatomic ideal gases. We build on this relation in Chapters 6–8, where we show how to compute changes in energy for any fluid at any temperature and density by systematically correcting the relatively simple ideal gas result. Let us explore more completely the assumptions of the ideal gas law by investigating the molecular origins of pressure. ig Pressure Pressure is the force exerted per unit area. We will be concerned primarily with the pressure exerted by the molecules of fluids upon the walls of their containers. Briefly, when molecules collide with the container walls, they must change momentum. The change in momentum creates a force on the wall. As temperature increases, the particles have more kinetic energy (and momentum) when they collide, so the pressure increases. We can understand this more fully with an ultrasimplified analysis of kinetic theory as it relates to the ideal gas law. Suppose we have two hard spherical molecules in a container that are bouncing back and forth with 1D velocity in the x-direction only and not contacting one another. We wish to quantify the forces acting on each wall. Since the particles are colliding only with the walls at A1 and A2 in our idealized motion, these are the only walls we need to consider. Let us assume that particles bounce off the wall with the same speed which they had before striking the wall, but in the opposite direction (a perfectly elastic collision where the wall is perfectly rigid and absorbs no momentum). A1 A2 Thus, the kinetic energy of the particles is fixed. If v is the initial velocity of the particle (recall that v is a vector quantity and notation v represents a scalar) before it strikes a wall, the change in velocity due to striking the wall is 2 v . The change in velocity of the particle indicates the presence of interacting forces between the wall and the particle. If we quantify the force necessary to change the velocity of the particle, we will also quantify the forces of the particle on the wall by Newton’s third principle. To quantify the force, we may apply Newton’s second principle stated in terms of momentum: The time rate of change of the momentum of a particle is equal to the resultant force acting on the particle and is in the direction of the resultant force. 8. We will formally define heat capacity and relations for CP and CV in Chapter 2. Section 1.4 Basic Concepts dp ------ = F dt 21 1.14 The application of this formula directly is somewhat problematic since the change in direction is instantaneous, and it might seem that the time scale is important. This can be avoided by determining the time-averaged force,9 F avg exerted on the wall during time t, tf tf dp F dt = Favg t = -----dt- dt = p ti 1.15 ti where p is the total change in momentum during time t. The momentum change for each collision is p = – 2mv where m is the mass per particle. Collision frequency can be related easily to the velocity. Each particle will collide with the wall every t seconds, where t 2 L /v, where L is the distance between A1 and A2. The average force is then v F avg = -----p- = – 2mv -----2L t 1.16 where v is the velocity before the collision with the wall. Pressure is the force per unit area, and the area of a wall is L2, thus m P = ------ v 12 + v 22 (1D ideal gas motion) L3 (ig) 1.17 where the subscripts denote the particles. If you are astute, you will recognize L3 as the volume of the box and the kinetic energy which we have shown earlier to relate to the temperature. If the particle motions are generalized to motion in arbitrary directions, collisions with additional walls in the analysis does not complicate the problem dramatically because each component of the velocity may be evaluated independently. To illustrate, consider a particle bouncing around the centers of four walls in a horizontal plane. From the top view, the trajectory would appear as below: y A2 A1 x For the same velocity as the first case, the force of each collision would be reduced because the particle strikes merely a glancing blow. The time of collisions between walls is now dependent on the component of velocity perpendicular to the walls. We have chosen a special case to illustrate here, where the box is square and the particle impacts at a 45° angle in the center of each wall. The x-component of the force can be related to the magnitude of the velocity by noting that vx vy, such that v (vx2 vy2)½ 9. See an introductory physics text for further discussion of time-averaged force. ! P is proportional to the number of particles in a volume and to the kinetic energy of the particles. 22 Unit I First and Second Laws vx2½. The time between collisions with wall A1 would be 4L/(v2½). The formula for the average force in two dimensions then becomes: v 2 v v 2 v mv 2 F avg,A = – 2mv x ---------- = – 2m ------- ---------- = – 2mv ------ = – ---------1 4L 2L L 2 4L 1.18 and the pressure due to two particles that do not collide with one another in two dimensions becomes: m P = --------- v 12 + v 22 2L 3 (2D motion) (ig) 1.19 (More complicated impact angles and locations will provide the same results but require more tedious derivations.) The extension to three dimensions is more difficult to visualize, but comparing Eqn. 1.17 to Eqn. 1.19, you should not be surprised to learn that the pressure in three dimensions is: m m P = --------- v 12 + v 2 2 = ------- v 12 + v 22 (3D motion) 3 3V 3L (ig) 1.20 The problem gets more complicated when collisions between particles occur. We ignored that possibility here because the ideal gases being considered are point masses that do not collide with one another. Including molecular collisions is a straightforward implementation of “billiard ball” physics. This subject is discussed further in Section 7.10 on page 276 and with great interactive graphics in the discontinuous molecular dynamics (DMD) module at Etomica.org. We see a relation developing between P and kinetic energy. When we insert the relation between temperature and kinetic energy (Eqn. 1.1) into Eqn. 1.20 we find that the ideal gas law results for a spherical (monatomic) molecule in 3D, ! Check your units when using this equation. 1J = 1kg-m2/s2. N nN A nM w 2 2 2 2 m N -----PV = v i V = ---- m v = ---------- m v = ----------- v = nRT (3D motion) 3V 3 3 3 i where m is the mass per particle and Mw is the molecular weight. A similar derivation with Eqn. 1.19 gives the results for motions restricted to 2D,10 m PV = ------ 2V 1.5 ! A 3D steam diagram is available as a MATLAB file called PVT.m. The diagram can be rotated to view the 2-D projections. (ig) 1.21 2 nN A m - v vi V = -------------2 2 nM w 2 = ----------- v = nRT 2D (2D motion) 2 (ig) 1.22 i REAL FLUIDS AND TABULATED PROPERTIES The thermodynamic behavior of real fluids differs from the behavior of ideal gases in most cases. Real fluids condense, evaporate, freeze, and melt. Characterization of the volume changes and energy changes of these processes is an important skill for the chemical engineer. Many real fluids do behave as if they are ideal gases at typical process conditions. P-V behavior of a very common real fluid (i.e., water) and an ideal gas can be compared in Figs. 1.3 and 1.4. Application of the 10. This is a pressure [=] force/area where motion is in 2-D and forces are in only two dimensions. In an alternative perspective molecules would only exist in a 2D plane. Then the divisor should be 2L2 and we multiply by area L2, and P2D [=] MPa-m, P2DL2 = nRT. Section 1.5 Real Fluids and Tabulated Properties 23 50 Temperature (oC) 40 Pressure (MPa) 100 30 Supercritical 200 Critical Point 300 400 20 Vapor Liquid 500 Sat’d Vapor 10 Sat’d Liquid 0 1 1.5 2 2.5 3 3.5 4 4.5 5 log10 [V (cm 3/mol)] Figure 1.4 P-V-T behavior of water at the same temperatures used in Fig. 1.3. The plot is prepared from the steam tables in Appendix E. ideal gas law simplifies many process calculations for common gases; for example, air at room temperature and pressures below 10 bars. However, you must always remember that the ideal gas law is an approximation (sometimes an excellent approximation) that must be applied carefully to any fluid. The behaviors are presented along isotherms (lines of constant temperature) and the deviations from the ideal gas law for water are obvious. Water is one of the most common substances that we work with, and water vapor behaves nearly as an ideal gas at 100C (Psat 0.1014 MPa), where experimentally the vapor volume is 1.6718 m3/kg (30,092 cm3/mol) and by the ideal gas law we may calculate V RT/P 8.314 · 373.15 / 0.1014 30,595 cm3/mol. However, the state is the normal boiling point, and we are well aware that a liquid phase can co-exist at this state. This is because there is another density of water at these conditions that is also stable.11 We will frequently find it convenient to work mathematically in terms of molar density or mass density, which is inversely related to molar volume or mass volume, 1/V. Plotting the isotherms in terms of density yields a P-diagram that qualitatively looks like the mirror image of the P-V diagram. Density is convenient to use because it always stays finite as P0, whereas V diverges. Examples of P- diagrams are shown in Fig. 7.1 on page 254. The conditions where two phases coexist are called saturation conditions. The terms “saturation pressure” and “saturation temperature” are used to refer to the state. The volume (or density) is called the saturated volume (or saturated density). Saturation conditions are shown in Fig. 1.4 as the “hump” on the diagram. The hump is called the phase envelope. Two phases coexist when the system conditions result in a state inside or on the envelope. The horizontal lines inside the curves are called tie lines that show the two volumes (saturated liquid and saturated vapor) that can coexist. The curve labeled “Sat’d Liquid” is also called the bubble line, since it represents conditions where boiling (bubbles) can occur in the liquid. The curve labeled “Sat’d Vapor” is also called a 11. This stability is determined by the Gibbs energy and we will defer proof until Chapter 9. ! Real fluids have saturation conditions, bubble points, and dew points. 24 Unit I First and Second Laws dew line, since it is the condition where droplets (dew) can occur in the vapor. Therefore, saturation is a term that can refer to either bubble or dew conditions. When the total volume of a system results in a system state on the saturated vapor line, only an infinitesimal quantity of liquid exists, and the state is indicated by the term “saturated vapor.” Likewise, when a system state is on the saturated liquid line, only an infinitesimal quantity of vapor exists, and the state is indicated by the term “saturated liquid.” When the total volume of the system results in a system in between the saturation vapor and saturation liquid volumes, the system will have vapor and liquid phases coexisting, each phase occupying a finite fraction of the overall system. Note that each isotherm has a unique saturation pressure. This pressure is known as the saturation pressure or vapor pressure. Although the vapor pressure is often used to characterize a pure liquid’s bubble point, recognize that it also represents the dew point for the pure vapor. ! Liquids are quite incompressible. Following an isotherm from the right side of the diagram along a path of decreasing volume, the isotherm starts in the vapor region, and the pressure rises as the vapor is isothermally compressed. As the volume reaches the saturation curve at the vapor pressure, a liquid phase begins to form. Notice that further volume decreases do not result in a pressure change until the system reaches the saturated liquid volume, after which further decreases in volume require extremely large pressure changes. Therefore, liquids are often treated as incompressible even though the isotherms really do have a finite rather than infinite slope. The accuracy of the incompressible assumption varies with the particular application. As we work problems involving processes, we need to use properties such as the internal energy of a fluid.12 Properties such as these are available for many common fluids in terms of a table or chart. For steam, both tables and charts are commonly used, and in this section we introduce the steam tables available in Appendix E. An online supplement is available to visualize the P-V and P-T representations in MATLAB permitting the user to interactively rotate the surface. Steam Tables When dealing with water, some conventions have developed for referring to the states which can be confusing if the terms are not clearly understood. Steam refers to a vapor state, and saturated steam is vapor at the dew point. For water, in the two-phase region, the term “wet steam” is used to indicate a vapor liquid system. ! The critical temperature and critical pressure are key characteristic properties of a fluid. Steam properties are divided into four tables. The first table presents saturation conditions indexed by temperature. This table is most convenient to use when the temperature is known. Each row lists the corresponding saturation values for pressure (vapor pressure), internal energy, volume, and two other properties we will use later in the text: enthalpy and entropy. Special columns represent the internal energy, enthalpy, and entropy of vaporization. These properties are tabulated for convenience, although they can be easily calculated by the difference between the saturated vapor value and the saturated liquid value. Notice that the vaporization values decrease as the saturation temperature and pressure increase. The vapor and liquid phases are becoming more similar as the saturation curve is followed to higher temperatures and pressures. At the critical point, the phases become identical. Notice in Fig. 1.4 that the two phases become identical at the highest temperature and pressure on the saturation curve, so this is the critical point. For a pure fluid, the critical temperature is the temperature at which vapor and liquid phases are identical on the saturation curve, and is given the notation Tc. The pressure at which this occurs is called the critical pressure, and is given the symbol Pc. A fluid above the critical temperature is often called supercritical. 12. Calculation of these properties requires mastery of several fundamental concepts as well as application of calculus and will be deferred. We calculate energies for ideal gas in Chapter 2 and for real fluids in Chapter 8. Section 1.5 Real Fluids and Tabulated Properties 25 The second steam table organizes saturation properties indexed by pressure, so it is easiest to use when the pressure is known. Like the temperature table, vaporization values are presented. The table duplicates the saturated temperature table, that is, plotting the saturated volumes from the two tables would result in the same curves. The third steam table is the largest portion of the steam tables, consisting of superheated steam values. Superheated steam is vapor above its saturation temperature at the given pressure. The adjective “superheated” specifies that the vapor is above the saturation temperature at the system pressure. The adjective is usually used only where necessary for clarity. The difference between the system temperature and the saturation temperature, (T T sat), is termed the degrees of superheat. The superheated steam tables are indexed by pressure and temperature. The saturation temperature is provided at the top of each pressure table so that the superheat may be quickly determined without referring to the saturation tables. The fourth steam table has liquid-phase property values at temperatures below the critical temperature and above each corresponding vapor pressure. Liquid at these states is sometimes called subcooled liquid to indicate that the temperature is below the saturation temperature for the specified pressure. Another common way to describe these states is to identify the system as compressed liquid, which indicates that the pressure is above the saturation pressure at the specified temperature. The adjectives “subcooled” and “compressed” are usually only used where necessary for clarity. Notice by scanning the table that pressure has a small effect on the volume and internal energy of liquid water. By looking at the saturation conditions together with the general behavior of Fig. 1.4 in our minds, we can determine the state of aggregation (vapor, liquid, or mixture) for a particular state. ! Superheat. ! Subcooled, compressed. Example 1.3 Introduction to steam tables For the following states, specify if water exists as vapor, liquid, or a mixture: (a) 110 C and 0.12 MPa; (b) 200C and 2 MPa; (c) 0.8926 MPa and 175C. Solution: (a) Looking at the saturation temperature table, the saturation pressure at 110 oC is 0.143 MPa. Below this pressure, water is vapor (steam). (b) From the saturation temperature table, the saturation pressure is 1.5549 MPa; therefore, water is liquid. (c) This is a saturation state listed in the saturation temperature table. The water exists as saturated liquid, saturated vapor, or a mixture. Linear Interpolation Since the information in the steam tables is tabular, we must interpolate to find values at states that are not listed. To interpolate, we assume the property we desire (e.g., volume, internal energy) varies linearly with the independent variables specified (e.g., pressure, temperature). The assumption of linearity is almost always an approximation, but is a close estimate if the interval of the calculation is small. Suppose we seek the value of volume, V, at pressure, P, and temperature, T, but the steam tables have only values of volume at P1 and P2 which straddle the desired pressure value as shown in Fig. 1.5. The two points represent values available in the tables and the solid line represents the true behavior. The dotted line represents a linear fit to the tabulated points. ! Linear interpolation is a necessary skill for problem solving using thermodynamic tables. 26 Unit I First and Second Laws Volume V1 V V2 P P1 P2 Pressure Figure 1.5 Illustration of linear interpolation. If we fit a linear segment to the tabulated points, the equation form is y mx b, where y is the dependent variable (volume in this case), x is the independent variable (pressure in this case), m is the slope m = y x = V 2 – V 1 P 2 – P 1 , and b is the intercept. We can interpolate to find V without directly determining the intercept. Since the point we desire to calculate is also on the line with slope m, it also satisfies the equation m = y x = V – V 1 P – P 1 . We can equate the two expressions for m to find the interpolated value of V at P. There are two quick ways to think about the interpolation. First, since the interpolation is linear, the fractional change in V relative to the volume interval is equal to the fractional change in P relative to the pressure interval. In terms of variables: P – P1 V – V1 ----------------- = ----------------V2 – V1 P2 – P1 For example, (V V1) is 10% of the volume interval (V2 V1), when (P P1) is 10% of (P2 P1). We can rearrange this expression to find: P – P1 V = V 1 + ------------------ V 2 – V 1 P –P 2 1.23 1 If we consider state “1” as the base state, we can think of this expression in words as V base V (fractional change in P) · (volume interval size) Another way to think of Eqn. 1.23 is by arranging it as: V2 – V1 V = V 1 + ------------------ P – P 1 P2 – P1 which in words is V base V slope · (change in P from base state) 1.24 27 Section 1.5 Real Fluids and Tabulated Properties Note that subscripts for 1 and 2 can be interchanged in any of the formulas if desired, provided that all subscripts are interchanged. In general, interpolation can be performed for any generic property M such that (modifying Eqn. 1.23) x – x1 M = M 1 + ---------------- M 2 – M 1 x –x 2 1.25 1 where M represents the property of interest (e.g., V) and x is the property you know (e.g., P). Example 1.4 Interpolation Find the volume and internal energy for water at: (a) 5 MPa and 325C and (b) 5 MPa and 269C. Solution: (a) Looking at the superheated steam table at 5 MPa, we find the saturation temperature in the column heading as 263.9C; therefore, the state is superheated. Values are available at 300C and 350C. Since we are halfway in the temperature interval, by interpolation the desired U and V will also be halfway in their respective intervals (which may be found by the average values): U (2699.0 2809.5)/2 2754.3 kJ/kg V (0.0453 0.0520)/2 0.0487 m3/kg (b) For this state, we are between the saturation temperature (263.9C) and 300C, and we apply the interpolation formula: 269 – 263.9 U = 2597.0 + ---------------------------- 2699.0 – 2597.0 = 2611.4 kJ/kg 300 – 263.9 3 269 – 263.9 V = 0.0394 + ---------------------------- 0.0453 – 0.0394 = 0.0402 m /kg 300 – 263.9 Double Interpolation Occasionally, we must perform double or multiple interpolation to find values. The following example illustrates these techniques. Example 1.5 Double interpolation For water at 160C and 0.12 MPa, find the internal energy. Solution: By looking at the saturation tables at 160C, water is below the saturation pressure, and will exist as superheated vapor, but superheated values at 0.12 MPa are not tabulated in the superheated table. If we tabulate the available values, we find 28 Unit I First and Second Laws Example 1.5 Double interpolation (Continued) 0.1 MPa 150 C 0.12 MPa 0.2 MPa 2582.9 2577.1 2658.2 2654.6 160 C 200 C We may either interpolate the first and third columns to find the values at 160C, followed by an interpolation in the second row at 160C, or interpolate the first and third rows, followed by the second column. The values found by the two techniques will not be identical because of the nonlinearities of the properties we are interpolating. Generally, the more precise interpolation should be done first, which is over the smaller change in U, which is the pressure interpolation. The pressure increment is 20% of the pressure interval [(0.12 0.1)/(0.2 0.1)]; therefore, interpolating in the first row, U 2582.9 0.2 · (2577.1 2582.9) 2581.7 kJ/kg and in the third row, U 2658.2 0.2 · (2654.6 2658.2) 2657.5 kJ/kg and then interpolating between these values, using the value at 150 C as the base value, U 2581.7 0.2 · (2657.5 2581.7) 2596.9 kJ/kg The final results are tabulated in the boldface cells in the following table: 150 C 0.1 MPa 0.12 MPa 0.2 MPa 2582.9 2581.7 2577.1 2658.2 2657.5 160 C 200C 2596.9 2654.5 We also may need to interpolate between values in different tables, like the saturated tables and superheated tables. This is also straightforward as shown in the following example. Example 1.6 Double interpolation using different tables Find the internal energy for water at 0.12 MPa and 110C. Solution: We found in Example 1.3 on page 25 that this is a superheated state. From the superheated table we can interpolate to find the internal energy at 110C and 0.1 MPa: U 2506.2 0.2 · (2582.9 2506.2) 2521.5 kJ/kg Section 1.5 Real Fluids and Tabulated Properties 29 Example 1.6 Double interpolation using different tables (Continued) At 0.2 MPa, 110C is not found in the superheated table because the saturation temperature is 120.3C, so the values at this pressure cannot be used. Therefore, we can find the desired internal energy by interpolation using the value above and the saturation value at 110C and 0.143 MPa from the saturation temperature table: 0.12 – 0.1 U 2521.5 --------------------------- (2517.7 2521.5) 2519.7 kJ/kg 0.143 – 0.1 Computer-Aided Interpolation Occasionally, interpolation must be performed when the T and P are both unknown. Computers or spreadsheets can be helpful as shown in the next example. Example 1.7 Double interpolation using Excel Steam undergoes a series of state changes and is at a final state where U 2650 kJ/kg and V 0.185 m3/kg. Find the T and P. Solution: Scanning the steam tables, the final state is in the range 1.0 MPa P 1.2 MPa, 200°C T 250°C. The final state requires a double interpolation using U and V. One easy method is to set up the table in Excel. In each of the tables below, the pressure interpolation is performed first in the top and bottom rows, dependent on the pressure variable in the top of the center column, which can be set at any intermediate pressure to start. The temperature interpolation is then entered in the center cell of each table using the temperature variable. The formulas in both tables reference a common temperature variable cell and a common pressure variable cell. Solver is started and T and P are adjusted to make U 2650 kJ/kg subject to the constraint V 0.185 m3/kg. (See Appendix A for Solver instructions.) The converged result is shown at T 219.6°C and P 1.17 MPa. Pf U(kJ/kg) table P = 1 MPa 1.164752 P = 1.2 MPa T = 200o C 2622.2 2614.539 2612.9 Tf 219.4486791 2650 T = 250o C 2710.4 2705.705 2704.7 P = 1.2 MPa V(m 3 /kg) table P = 1 MPa T = 200o C 0.2060 0.175768 0.1693 0.185 0.2327 0.199502 0.1924 T = 250o C Extrapolation Occasionally, the values we seek are not conveniently between points in the table and we can apply the “interpolation” formulas to extrapolate as shown in Fig. 1.6. In this case, T lies outside the interval. Extrapolation is much less certain than interpolation since we frequently do not know “what Example use of Excel for double interpolation. 30 Unit I First and Second Laws Internal Energy curve lies beyond” that we may miss by linear approximation. The formulas used for extrapolation are identical to those used for interpolation.With the steam tables, extrapolation is generally not necessary at normal process conditions and should be avoided if possible. U U2 U1 T1 T2 T Pressure Figure 1.6 Illustration of linear extrapolation. Phase Equilibrium and Quality Along the saturation curve in Fig. 1.4 on page 23, there is just one degree of freedom (F C P 2 1 2 2 1). If we seek saturation, we may choose either a T sat or a P sat, and the other is determined. The vapor pressure increases rapidly with temperature as shown in Fig. 1.7. A plot of ln P sat sat versus 1 T is nearly linear and over large intervals, so for accurate interpolations, vapor pressure data should be converted to this form before interpolation. However, the steam tables used Psat Liquid Solid Critical Point Solid Liquid r d ui Vapor Triple Point So lid Va po r po Va q Li T sat Figure 1.7 P-T representation of real fluid behavior. Note that only vapor and liquid behavior is shown in Fig. 1.4 on page 23. Section 1.5 Real Fluids and Tabulated Properties 31 with this text have small enough intervals that direct interpolation can be applied to P sat and T sat without appreciable error. The saturation volume values of the steam tables were used to generate the phase diagram of Fig. 1.4 on page 23. Note that as the critical point is approached, the saturation vapor and liquid values approach one another. The same occurs for internal energy and two properties that will be used in upcoming chapters, enthalpy, H, and entropy, S. When a mixture of two phases exists, we must characterize the fraction that is vapor, since the vapor and liquid property values differ significantly. The mass percentage that is vapor is called the quality and given the symbol q. The properties V, U, H, and S, may be represented with a generic variable M. The overall value of the state variable M is M (1 q) ML qMV 1.26 which may be rearranged as M ML q(MV ML) but (MV ML) is just Mvap and for internal energy, enthalpy, and entropy, it is tabulated in columns of the saturation tables. The value of overall M is M M L qM vap 1.27 Look carefully at Eqn. 1.27 in comparison with Eqn. 1.25; it may be helpful to recognize a quality calculation as an interpolation between saturated liquid and saturated vapor. Two examples help demonstrate the importance of quality calculations. Example 1.8 Quality calculations Two kg of water coexists as vapor and liquid at 280C in a 0.05 m3 rigid container. What is the pressure, quality, and overall internal energy of the mixture? Solution: The overall mass volume is V 0.05 m3/ 2 kg 0.025 m3/ kg. From the saturation temperature table, the pressure is 6.417 MPa. Using the saturation volumes at this condition to find q, 0.025 0.001333 q (0.0302 0.0013) m3/ kg which leads to q 0.82. The overall internal energy is U 1228.33 0.82 · 1358.1 2342 kJ/ kg ! Quality is the vapor mass percentage of a vapor/ liquid mixture. 32 Unit I First and Second Laws Example 1.9 Constant volume cooling Steam is initially contained in a rigid cylinder at P 30 MPa and V 102.498cm3/mole. The cylinder is allowed to cool to 300C. What is the pressure, quality, and overall internal energy of the final mixture? Solution: The overall mass volume is V 102.498cm3-mole1 · 106(m3/cm3)/(18.02E-3kg/mole) 0.01747 m3/kg. From the superheated steam table at 30 MPa, the initial temperature is 900C. When the cylinder is cooled to 300C, the path is shown in Fig. 1.8 below. You should notice that there is no pressure in the superheated steam tables that provides a volume of V 0.01747 m3/kg. Look hard, they are all too large. (Imagine yourself looking for this on a test when you are in a hurry.) Now look in the saturated steam tables at 300C. Notice that the saturated vapor volume is 0.0217 m3/ kg. Since that is higher than the desired volume, but it is the lowest vapor volume at this temperature, we must conclude that our condition is somewhere between the saturated liquid and the saturated vapor at a pressure of 8.588 MPa. (When you are in a hurry, it is advisable to check the saturated tables first.) Using the saturation volumes at 300C condition to find q, 0.01747 0.001404 q (0.0217 0.001404) m3/kg which leads to q (0.01747 0.001404)/(0.0217 0.001404) 0.792. The overall internal energy is U 1332.95 0.792 · 1230.67 2308 kJ/kg 50 Temperature (oC) 40 Pressure (MPa) 100 200 30 300 400 20 500 10 0 1 1.5 2 2.5 3 3.5 3 log10 [V (cm /mol)] Figure 1.8 P-V-T behavior of water illustrating a quality calculation. 4 4.5 5 Section 1.6 Summary 1.6 33 SUMMARY Years from now you may have some difficulty recalling the details presented in this text. On the other hand, the two basic premises outlined in this introductory section are so fundamental to technically educated people that you really should commit them to long-term memory as soon as possible. Formally, we may state our two basic premises as the first and second “laws” of thermodynamics.13 First Law: Overall energy is conserved (you can’t get something for nothing). Second Law: Overall entropy changes are greater than or equal to zero (generation of disorder results in lost work). The first law is further developed in Chapters 2 and 3. The concepts of entropy and the second law are developed in Chapter 4 and process applications in Chapter 5. The exact relationship between the two basic premises and these two laws may not become apparent until some time later in this text, but you should begin to absorb and contemplate these fundamentals now. There are times when the endeavor to apply these simple laws seems daunting, but the answer appears simple in retrospect, once obtained. By practicing adaptation of the basic principles to many specific problems, you slowly grasp the appropriate connection between the basic premises and finding the details. Try not to be distracted by the vocabulary or the tedious notation that goes into keeping all the coupled systems classified in textbook fashion. Keep in mind that other students have passed through this and found the detailed analysis to be worth the effort. Important Equations The content of this chapter is primarily about laying down the fundamental concepts more than deriving equations. Nevertheless, three concepts that we refer to repeatedly can be expressed by equations. Of course, the ideal gas law is important, but an implication of it that may be new is: Mw 2 T = -------- v (for monatomic molecules in 3D) 3R 1.28 This equation conveys that temperature is closely related to molecular kinetic energy. Although derived with the ideal gas assumption, it turns out to be true for real fluids as well. Every time you alter the temperature, you should think about the implications for molecular kinetic energy. Another important equation relates to deviations from the ideal gas law: if r u r = – if r 0 if r 1.29 This is the square well potential model, the simplest characterization of how real molecules attract and repel. As you add energy to the system, real fluids may absorb that energy by moving molecules from inside the square well to outside, converting potential energy into kinetic energy without altering the temperature as an ideal gas would. A simple example is boiling. Ideal gases cannot 13. There is also a third law of thermodynamics, as discussed by Denbigh, K., 1981. The Principles of Chemical Equilibrium, London: Cambridge University Press, p. 416. The third law is of less direct interest in this introductory text, however. 34 Unit I First and Second Laws boil, but real fluids can. This interplay between kinetic energy, temperature, and potential energy pervades many discussions throughout the text. Finally, we can write a generic equation that symbolizes the procedure for interpolation: x – x1 M = M 1 + ---------------- M 2 – M 1 x –x 2 1.30 1 A similar equation is used for quality calculations which can be viewed as an interpolation between saturated liquid and saturated vapor. Throughout Unit I, we refer extensively to the steam tables and interpolation to account for deviations from the ideal gas law. Test Yourself 1. Draw a sketch of the force model implied by the square-well potential, indicating the position(s) where the force between two atoms is zero and the positions where it is nonzero. 2. Explain in words how the pressure of a fluid against the walls of its container is related to the velocity of the molecules. 3. What is it about molecules that requires us to add heat to convert liquids to gases? 4. If the kinetic energy of pure liquid and vapor molecules at phase equilibrium must be the same, and the internal energy of a system is the sum of the kinetic and potential energies, what does this say about the intensive internal energy of a liquid phase compared with the intensive internal energy of the gas phase? 5. Explain the terms “energy,” “potential energy,” “kinetic energy,” and “internal energy.” 6. How is the internal energy of a substance related to the intermolecular pair potentials of the molecules? 7. Are T and P intensive properties? Name two intensive properties and two extensive properties. 8. How many degrees of freedom exist when a pure substance coexists as a liquid and gas? 9. Can an ideal gas condense? Can real fluids that follow the ideal gas law condense? 10. Give examples of bubble, dew, saturation, and superheated conditions. Explain what is meant when wet steam has a quality of 25%. 11. Create and solve a problem that requires double interpolation. 1.7 PRACTICE PROBLEMS P1.1 Estimate the average speed (mph) of hydrogen molecules at 200 K and 3 bars. ANS. 3532 P1.2 Estimate the entropy (J/g-K) of steam at 27.5MPa and 425C. ANS. 5.1847 Section 1.8 Homework Problems 1.8 35 HOMEWORK PROBLEMS Note: Some of the steam table homework problems involve enthalpy, H, which is defined for convenience using properties discussed in this chapter, H U + PV. The enthalpy calculations can be performed by reading the tabulated enthalpy values from the tables in an analogous manner used for internal energy. We expect that students will be introduced to this property in course lectures in parallel with the homework problems that utilize H. 1.1 In each of the following, sketch your estimates of the intermolecular potentials between the given molecules and their mixture on the same pair of axes. (a) Chloroform is about 20% larger than acetone and about 10% stickier, but chloroform and acetone stick to one another much more strongly than they stick to themselves. (b) You have probably heard that “oil and water don’t mix.” What does that mean in molecular terms? Let’s assume that oil can be characterized as benzene and that benzene is four times larger than water, but water is 10% stickier than benzene. If the 12 parameter is practically zero, that would represent that the benzene and water stick to themselves more strongly than to one another. Sketch this. 1.2 For each of the states below, calculate the number of moles of ideal gas held in a three liter container. (a) T = 673 K, P = 2 MPa (b) T = 500 K, P = 0.7 MPa (c) T = 450 K, P = 1.5 MPa 1.3 A 5 m3 outdoor gas storage tank warms from 10°C to 40°C on a sunny day. If the initial pressure was 0.12 MPa at 10°C, what is the pressure at 40°C, and how many moles of gas are in the tank? Use the ideal gas law. 1.4 An automobile tire has a pressure of 255 kPa (gauge) in the summer when the tire temperature after driving is 50 C. What is the wintertime pressure of the same tire at 0C if the volume of the tire is considered the same and there are no leaks in the tire? 1.5 A 5 m3 gas storage tank contains methane. The initial temperature and pressure are P = 1 bar, T = 18°C. Using the ideal gas law, calculate the P following each of the successive steps. (a) (b) (c) (d) 1 m3 (at standard conditions) is withdrawn isothermally. The sun warms the tank to 40°C. 1.2 m3 (at standard conditions) is added to the tank and the final temperature is 35°C. The tank cools overnight to 18°C. 1.6 Calculate the mass density of the following gases at 298 K and 1 bar. (a) Nitrogen (b) Oxygen (c) Air (use average molecular weight) (d) CO2 (e) Argon 36 Unit I First and Second Laws 1.7 Calculate the mass of air (in kg) that is contained in a classroom that is 12m x 7m x 3m at 293 K and 0.1 MPa. 1.8 Five grams of the specified pure solvent is placed in a variable volume piston. What is the volume of the pure system when 50% and 75% have been evaporated at: (i) 30°C, (ii) 50 C? Use the Antoine equation (Appendix E) to relate the saturation temperature and saturation pressure. Use the ideal gas law to model the vapor phase. Show that the volume of the system occupied by liquid is negligible compared to the volume occupied by vapor. (a) (b) (c) (d) (e) Hexane (L = 0.66 g/cm3) Benzene (L = 0.88 g/cm3) Ethanol (L = 0.79 g/cm3) Water without using the steam tables (L = 1 g/cm3) Water using the steam tables 1.9 A gasoline spill is approximately 4 liters of liquid. What volume of vapor is created at 1 bar and 293 when the liquid evaporates? The density of regular gasoline can be estimated by treating it as pure isooctane (2,2,4-trimethylpentane L 0.692 g/cm3) at 298 K and 1 bar. 1.10 The gross lifting force of a balloon is given by (air gas)Vballoon. What is the gross lifting force (in kg) of a hot air balloon of volume 1.5E6 L, if the balloon contains gas at 100°C and 1 atm? The hot gas is assumed to have an average molecular weight of 32 due to carbon dioxide from combustion. The surrounding air has an average molecular weight of 29 and is at 25 °C and 1 atm. 1.11 LPG is a useful fuel in rural locations without natural gas pipelines. A leak during the filling of a tank can be extremely dangerous because the vapor is denser than air and drifts to low elevations before dispersing, creating an explosion hazard. What volume of vapor is created by a leak of 40L of LPG? Model the liquid before leaking as propane with L 0.24 g/cm3. What is the mass density of pure vapor propane after depressurization to 293 K and 1 bar? Compare with the mass density of air at the same conditions. 1.12 The gas phase reaction A 2R is conducted in a 0.1 m3 spherical tank. The initial temperature and pressure in the tank are 0.05 MPa and 400 K. After species A is 50% reacted, the temperature has fallen to 350 K. What is the pressure in the vessel? 1.13 A gas stream entering an absorber is 20 mol% CO2 and 80 mol% air. The flowrate is 1 m3/min at 1 bar and 360 K. When the gas stream exits the absorber, 98% of the incoming CO2 has been absorbed into a flowing liquid amine stream. (a) What are the gas stream mass flowrates on the inlet and outlets in g/min? (b) What is the volumetric flowrate on the gas outlet of the absorber if the stream is at 320 K and 1 bar? 1.14 A permeation membrane separates an inlet air stream, F, (79 mol% N2, 21 mol% O2), into a permeate stream, M, and a reject stream, J. The inlet stream conditions are 293 K, 0.5 MPa, and 2 mol/min; the conditions for both outlet streams are 293 K and 0.1 MPa. If the permeate stream is 50 mol% O2, and the reject stream is 13 mol% O2, what are the volumetric flowrates (L/min) of the two outlet streams? Section 1.8 Homework Problems 37 1.15 (a) What size vessel holds 2 kg water at 80°C such that 70% is vapor? What are the pressure and internal energy? (b) A 1.6 m3 vessel holds 2 kg water at 0.2 MPa. What are the quality, temperature, and internal energy? 1.16 For water at each of the following states, determine the internal energy and enthalpy using the steam tables. T(°C) P(MPa) (a) 100 0.01 (b) 550 6.25 (c) 475 7.5 (d) 180 0.7 1.17 Determine the temperature, volume, and quality for one kg water under the following conditions: (a) U = 3000 kJ/kg, P = 0.3 MPa (b) U = 2900 kJ/kg, P = 1.7 MPa (c) U = 2500 kJ/kg, P = 0.3 MPa (d) U = 350 kJ/kg, P = 0.03 MPa 1.18 Two kg of water exist initially as a vapor and liquid at 90°C in a rigid container of volume 2.42 m3. (a) At what pressure is the system? (b) What is the quality of the system? (c) The temperature of the container is raised to 100°C. What is the quality of the system, and what is the pressure? What are H and U at this point relative to the initial state? (d) As the temperature is increased, at what temperature and pressure does the container contain only saturated vapor? What is H and U at this point relative to the initial state? (e) Make a qualitative sketch of parts (a) through (d) on a P-V diagram, showing the phase envelope. 1.19 Three kg of saturated liquid water are to be evaporated at 60°C. (a) At what pressure will this occur at equilibrium? (b) What is the initial volume? (c) What is the system volume when 2 kg have been evaporated? At this point, what is U relative to the initial state? (d) What are H and U relative to the initial state for the process when all three kg have been evaporated? (e) Make a qualitative sketch of parts (b) through (d) on a P-V diagram, showing the phase envelope. INDEX A absolute temperature, 7 acentric factor, 254 acid, 704 acidity parameter, 421 activity, 428, 646, 667, 669, 739 coefficient, 413, 425, 428, 759 pressure dependence, 440 rational, 759 temperature dependence, 441 unsymmetric convention, 446 adiabat, 150 adiabatic, xxi, 46, 74, 148 reaction temperature, 664, 813 adiabatic compressibility, 246, 249 adiabatic flash, 372, 380 aggregation state, 25 Amagat’s law, 106 amino acid, 719 anabolic reaction, 182, 734 anode, 727 anti-solvent, 570 Antoine equation, 62, 342, 847 See also vapor pressure apparent basis, 699 mole fraction, 772 moles, 792, 793 approach temperature, 100 association, xxi, 767 athermal, 477 ATP, 734 auxiliary property, 227 availability, 177, 672, 732 Avogadro’s number, xxv, 9 azeotrope, xxi, 393, 394, 418, 442, 473, 616, 630 double, 432 B Bancroft point, 432, 497 barotropy, xxi basicity parameter, 421 binary interaction parameter, 13, 468, 580, 582 binary vapor cycle, 211 binding polynomial, 733 binodal, xxi biological systems, 119 blood, 453 boiler, 70, 201 boilup ratio, 101 Boltzmann’s constant, 9 bottoms, 101 bottoms product, 101 boundaries, 74, 85 Boyle temperature, 292 bubble line, 23, 371, 394 bubble point pressure, 371, 376, 413, 416, 417, 586, 594, 832, 835 temperature, 371, 378, 381, 500, 549, 835 buffer, 723 C C-mole, 732 carbohydrate, 119 carboxylic acid, 627 carboxylic acid, ideal gas deviations, 424, 769 Carnahan-Starling, See equation of state Carnot cycle, 96, 144, 158, 199 869 870 Index Carnot heat pump, 99 Carnot, Sadi, 96 cascade refrigeration or cascade vapor cycle, 211 catabolic reaction, 182, 734 cathode, 727 cell (battery), 727 chain rule, 231 Chao-Seader model, 447 charge balance, 698, 704 charge-transfer complexes, 770 chemical potential, 398, 400, 427, 582, 739 ideal gas component, 402 Chemical Safety Board, 389 chemical-physical theory, 779 Clapeyron equation, 337, 338 Clausius, Rudolf, 130, 131, 233 Clausius-Clapeyron equation, 62, 337, 338 closed system, 16 coefficient of performance, 99, 158 coefficient of thermal expansion, 236 cohesive energy density, 472 coke formation, 670 colligative property, 453, 695 combinatorial contribution, 132, 506 combining rule, 13, 580, 582 compressed liquid, 25, 54, 55, 60 compressibility See also adiabatic compressibility, isothermal compressibility compressibility factor, 253 compressor, 73, 163, 173 multistage, 174 concentration gradient, 44 concurrent, 70 condenser, 70 configurational energy, 12, 132, 282 configurational entropy, 132 conjugate acid, 705, 709 conjugate base, 705, 708 conjugate pairs, 228 consistency thermodynamic, 440, 441 constant molar overflow, 101 contraction/expansion work, 86 See also expansion/contraction work control volume, 16 convenience property, 227 conversion, 110, 644 COP, 99 copolymer, 479 corresponding states, 253, 580 COSMO-RS, 520 countercurrent, 70 cricondenbar, 601 cricondentherm, 601 critical locus, 601, 614 critical point, 24, 253, 261, 262, 270, 599, 602, 614, 824 critical pressure, 24, 253 critical temperature, 24, 253 cubic equation, 260 solutions, 265, 822 D dead state, xxi, 178 deaerator, 206 Debye, 142 Debye-Hückel, 741 degree of reduction, 731 degrees of freedom, 18, 30, 59, 241, 370 degrees of superheat, 25 delta notation, 53 density, 17 density functional theory, 521 departure function, 115, 301, 303 deviations from Henry’s law, 444 deviations from Raoult’s law, 394, 444 dew line, 24, 371, 394 dew point, 372, 812 pressure, 377, 833, 836 temperature, 371, 378, 381, 488, 833, 837 diathermal, xxi, 46 dielectric constant, 741, 848 differentiation, 820 diffusion, 5, 44, 176 coefficient, 5 dipole, 8, 9 dispersion force, 10 distillate, 101 distillation column, 101 DNA, 719, 725 double interpolation, 27 downcomer, 102 Dulong and Petit, 59 duty, 101, 209 E economizer, 211 efficiency, xxi thermal, 96 turbine and compressor, 164 Einstein solid model, 139, 240 Einstein, Albert, 139, 239 electrolyte, 693, 813 electroneutrality, 704 electrostatic force, 9 elementary reaction, 671 endergonic, 652, 662, 728 Index endothermic, 112, 119, 427, 667 energy, 6 configurational, 12, 282 of fusion, 63 of vaporization, 24, 62 See also potential energy, kinetic energy, internal energy energy balance, 39 closed-system, 48 complete, 56 hints, 74 reaction, 113 reactions, 664 steady-state, 51 energy equation, 283 energy of mixing, 105, 106 engines, 214 enthalpy, 31, 35, 52, 57, 227 excess, See enthalpy of mixing, excess enthalpy of formation, 113, 843 of fusion, 63, 148, 559, 846 of mixing, 105, 106, 107, 113, 427 of reaction (standard state), 112 of vaporization, 24, 62, 101, 846 See also latent heat entity, electrolyte, 699 entrainer, 486, 490 entropy, 5, 31, 129 and heat capacity, 150, 229, 242 combinatorial, 512 configurational, 132 excess, See excess entropy generation, 142, 164, 672 ideal gas mixture, 139 ideal solution, 139 macroscopic, 142 microscopic, 132 of fusion, 148, 559 of mixing, 105, 138, 401, 402 of vaporization, 24, 148 thermal, 132, 139 entropy balance, 153 hints, 185 Environmental Protection Agency, 386 enzyme, 719 EOS, xxi, 260 EPA, 386 equal area rule, 357 equation of state, 18, 74, 235, 251, 347, 351, 353, 579 Benedict-Webb-Rubin, 260 Carnahan-Starling, 280, 294, 357 cubic solution, 263 ESD, 280, 295, 327, 357, 802 871 HR-SAFT, 800 Lee-Kesler, 260, 293 MCSL, 800 PC-SAFT, 327, 800 Peng-Robinson, 262, 266, 314, 316, 320, 347, 354, 357, 585, 587, 592, 594, 599, 601, 813 Redlich-Kwong, 325, 357 SAFT, 799, 800 Scott, 280, 294 Soave-Redlich-Kwong, 294, 327 stable roots, 268 van der Waals, 261, 271, 280, 355, 591 virial, 258, 288, 313, 347, 580, 590, 813 equilibrium, 5 criteria chemical reaction, 644, 813 liquid-liquid, 540 liquid-vapor, 336, 399, 401 Raoult’s law, 404 definition, 16 liquid-liquid, 539, 614, 624 liquid-liquid-vapor, 540, 614 solid-liquid-vapor, 624 equilibrium constant, 646 ideal gases, 642 ESD, See equation of state Euler’s reciprocity relation, 233 eutectic, 561, 562, 627 exact differential, 232 Excel, 813 excess enthalpy, 426, 440 entropy, 440 Gibbs energy, 413, 427, 440 properties, 426 volume, 426, 440, 581 exergonic, 652, 662, 728 exergy, 177 exergy, See availability exothermic, 112, 119, 427 exothermic reaction, 655, 666 expander, 72 See also turbine expansion rule, 231 expansion/contraction work, 40, 86 extensive properties, 18, 398 extent of reaction, 110 extractive distillation, 392, 490 extrapolation, linear, 29 F FAD, 735 Faraday’s constant, 728 872 Index fat, 119 feedwater preheater, 205 Fenske equation, 393 first law of thermodynamics, 4, 39 flash, 371 adiabatic, 372 drum, 378 isothermal, 372, 378, 546, 596, 812, 834, 838 multicomponent, 381 point, 389, 535 flavin adenine dinucleotide, 735 Flory equation, 474, 491, 506, 510, 514 Flory-Huggins theory, 474, 491, 541 force frictional, 86 from potential energy, 8 formal concentration, 699 Fourier’s law, 157 free energies, 228 free volume, 475 friction, 43 friction factor, 68 fugacity, xxi, 268, 344, 400, 804 coefficient, 345, 425, 583, 585 component by EOS, 584 ideal gas component, 403 ideal gas pure, 347 liquid, 351 liquid component, 425 solid, 353 fundamental property relation, 226 fusion, 63, 559 G gamma-phi method, 425 gas constant, 9 generalized correlation, 256, 323 Gibbs energy, 178, 228, 268 excess, See excess Gibbs energy of a mixture, 427 of formation, 647, 843 of fusion, 558 of mixing, 105, 403, 428 of reaction (standard state), 647 Gibbs phase rule, 18, 229, 370, 617 Gibbs, J.W., 178 Gibbs-Duhem equation, 439, 440, 755 Gibbs-Helmholtz relation, 441, 653, 746 global warming factor, 497 gradient concentration, 44 pressure, 44 temperature, 44 velocity, 44 Grayson-Streed model, 447 group contributions, 511 H hard-sphere fluid, 285 head space, 387 heat, 6, 17, 46 of formation (standard state), 112 of fusion, See latent heat, enthalpy of fusion of mixing, 107, 115, 427 of reaction, 60, 111 of reaction (standard state), 112 of vaporization, See latent heat, enthalpy of vaporization sign convention, 53 heat capacity, 19, 57, 235, 244, 269, 843 and entropy, 149, 240, 242 heat conduction, 17, 157, 176 heat convection, 17 heat engine, 157 See also Carnot cycle heat exchanger, 70, 160 heat radiation, 17 heavy key, 392, 487 Helmholtz energy, 179, 228, 309, 584 Helmholtz energy of mixing, 105 Helmholtz, Hermann, 179 Henderson-Hasselbalch, 724 Henry’s law, 376, 443, 457, 535, 610, 740, 749 standard state, 759 Hess’s law, 111, 119, 648 heteroazeotrope, xxi, 550, 616, 630, 633 Hilsch vortex tube, 195 Hooke’s law, 139 human blood, 453 hydrate inhibitor, 603 hydration, 694 hydrogen bonding, 8, 466, 501, 533, 550, 603, 769 hydronium, 704 hydrophilic, 421 hydrophobic, 421 hydrophobic effect, 485 hypertonic, 453 I ideal chemical theory, 776, 813 ideal gas law, 18, 284 ideal solution, 106, 139, 403 incompressible fluid, 24 induction forces, 10 inelastic collision, 44 inert, 387, 650 infinite dilution, xxi, 434, 471 instability, 539 Index integration, 820 intensive, 17 internal energy, 15, 57, 226 ideal gas, 19 interpolation, 25 double, 27 interstage cooling, 174, 211 ion pair, 704 ion pair chromatography, 704 irreversible, xxi, 43, 74, 86, 142 isenthalpic, xxi isentropic, xxi, 143, 148 isentropic efficiency, xxi isobaric, xxi, 46, 147 isobaric coefficient of thermal expansion, 236 isochore, xxii, 46, 147 isochoric, 274 isoelectric point, 724 isolated, xxii, 74 system, 16, 50 isopiestic, xxii isopleth, 601 isopycnic, xxii isosteric, xxii isotherm, 23, 264 isothermal, xxii, 46, 147 compressibility, 236, 253, 824 flash, 372, 834, 838 isotonic, 453 IUPAC, 728 J Jacobian, 242, 244 Joule-Thomson coefficient, 212, 236, 245 Joule-Thomson expansion, 68 K Kamlet-Taft acidity/basicity, 421, 771 kinetic energy, 6, 22, 43, 66, 69 K-ratio, 372, 586 LLE, 545 modified Raoult’s law, 425 Raoult’s law, 376 shortcut, 376 SLE, 566 L latent heat, 60, 846 laws See Raoult’s law, Henry’s law See zeroth law, first law, second law, third law Le Châtelier, Henry, 649 Le Châtelier’s principle, 649, 699 Lee-Kesler, See equation of state Legendre transformation, 228, 244, 398, 745 leveling effect, 705 lever rule, 372, 543, 551, 571, 630 Lewis, G.N., 3, 355 Lewis-Randall rule, 404, 443, 739 LFL, lower flammability limit, 389 light key, 392, 487 Linde liquefaction, 212, 320 linear extrapolation, 29 liquefaction, 212, 320 LLE, xxii, 539 local composition, 501 Lorentz and Bertholet combining rule, 13 lost work, 15, 42, 137, 142, 164, 178 low pressure steam, 172 M M, as a generic property, 31, 303, 398, 426 MAB, Margules acid-base model, 421 Margules, 435, 812, 813 acid-base characterization, 421 one-parameter, 414, 484, 509, 542 two-parameter, 430 mass balance, 16 master equation, xxii matrix, 817 maximum boiling azeotrope, 397 maximum pressure azeotrope, 394 Maxwell’s relations, 233 McCabe-Thiele analysis, 391 McMillan-Mayer framework, 450 mean ionic activity coefficients, 753 mean molality, 753 measurable properties, xxii, 229, 235 metastable, xxii, 269 See also unstable, equation of state (stable roots) micelle, 691 microstate, 132 minimum boiling azeotrope, 394 minimum pressure azeotrope, 397 mixing rule, 466, 580, 840 differentiation, 588 molecular basis, 466 mixture combining rule, 13 modified Raoult’s law, 412, 427 molal standard state, 759, 849 molality, 700 molecular asymmetry, 619 molecular simulation, 276 Mollier diagram, 166 monomer, 772 MOSCED, 480 873 874 Index mutual solubilities, 423, 540, 544 N NAD, 735 negative deviations, 413, 444 Nernst equation, 728 Newton-Raphson, 265 nicotinamide adenine dinucleotide, 735 nominal, 699 noncondensable, 387, 487 normal boiling temperature, 61 normalization, 488, 595, 833 nozzle, xxii, 69, 74, 162 NRTL model, 508 O one-root region, 264 open system, 16, 51 osmotic coefficient, 742 osmotic pressure, 449, 743, 755 osmotic virial coefficient, 451 overall molar volume, enthalpy, etc., 31 mole fraction, 371 overall property, 31 overhead product, 101 oxidation, 727 oxidation state, 729 P packing fraction, 273 packing, column, 102 parameter binary interaction, 13, 582 cross, 467 parity plot, 578 partial condenser, 101, 378 partial molar excess volume, 426 Gibbs energy, 398, 400, 427 properties, 398 volume, 398, 426 partial pressure, 403, 424 path properties, 46 PC-SAFT, See equation of state Peng-Robinson, See equation of state permutations, 135 pH, 701 phase behavior classes, 617 phase envelope, 23, 370, 601, 617 photosynthesis, 182 pI, 724 Pitzer correlation, 256, 323, 349 Plait point, 552 pMg, 736 polytropic, xxii positive deviations, 413, 444 potential energy, 8, 66 hard sphere, 277 Lennard-Jones, 10 square-well, 10, 274, 829 Sutherland, 10 potential energy, 6 Poynting correction, 350, 353, 425, 667, 669 Prausnitz-Shair, 447 preheater, 205 pressure, 20 bubble, See bubble-point pressure dew, See dew-point pressure equation, 283 gradient, 44 reduced reference, 63 See also reference state probability, 133, 467 process simulators, 839 properties convenience, 227 measurable, 229 protein, 119, 719 protein engineering, 479 proton condition, 706 pseudoisomer, 736 pump, 73, 163, 173, 202 purge gas, 387 P-x-y diagram, 370 Q quadratic equation, 817 quadratic mixing rule, 467 quadrupole, 8 quality, xxii, 30, 167, 336 R Rackett equation, 351 radial distribution function, 284, 828 Rankine cycle, 200 Raoult, François-Marie, 376, 695 Raoult’s law, 405, 443 modified, 412, 427 negative deviations, 394, 444, 781 positive deviations, 394, 444, 781 rdf, See radial distribution function reaction coordinate, 110, 643 elementary, 671 reboiler, 101 Index reciprocity relation, Euler, 233 rectifying section, 101 Redlich-Kister activity coefficient model, 429 Redlich-Kwong, See equation of state, 325 redox reactions, 727 reduced, 254 pressure, See pressure, reduced temperature, See temperature, reduced reduction, 727 reference state, xxii, 6, 63, 112, 113, 314, 318, 652 reflux, 101 ratio, 101, 393 refrigerant choice, 210 refrigeration, 208 regular solution theory, 468 See also van Laar, Scatchard-Hildebrand theory relative volatility, 392, 394, 456, 490 repulsive force, 10 reservoir, 17 residual contribution, 506 residue curve, 630, 813 retrograde condensation, 601 reverse osmosis, 449 reversible, 42, 74, 142 internally, 158 work, 41 roots See quadratic, cubic three- and one-root regions, 264 rotational energy, 43, 59 rules of thumb plotting on logarithmic coordinates, 712 S SAFT, See equation of state salting in, out, 725 saturated liquid, 23 steam, 24 saturation, 23 pressure, 23 temperature, 23 scalar, 20 Scatchard-Hildebrand, 471, 812 Schrödinger equation, 8 Scott, See equation of state second law of thermodynamics, 4, 130, 131, 143 sensible heat, xxii, 60, 111 separation of variables, 75, 77 separatrices, 631 separatrix, 631, 635 shortcut distillation calculation, 393, 442, 630 sign convention heat and work, 53 mass balance, 16 work, 40 simple system, 131, 226 sink, 17 SLE, xxii, 556 Soave, See equation of state solubility, 540, 560, 726 parameter, 472, 621 solubility product, 726 solvation, xxii, 521, 694, 767 speciation, 697 specific heat, xxii specific property, xxii, 17 spinodal, xxii, 269, 553 split, 392 SSCED model, 482 stability check, 358 stable roots, 268 stage, separation, 102 standard conditions, xxii standard hydrogen electrode, 728 standard state, xxii, 112, 425, 428, 645 Gibbs energy of reaction, 648 heat of formation, 112 heat of reaction, 111 Henry’s law, 759 molal, 759, 849 state, 18 of aggregation, xxii, 25, 60, 63, 113, 318 property, 46 reference, 63, 318 variable, 18 statistical thermodynamics, 59 steady-state, xxiii, 18, 74 energy balance, 51 steam properties, 811 steam tables, 24, 854 steam trap, 92 Stirling's approximation, 136 stoichiometric coefficient, 109 stoichiometric number, 109 stoichiometry, 109 STP, xxiii strategy problem-solving, 74, 165, 177, 179, 373 stripping section, 101 strong acid, 704 strong base, 704 strong electrolyte, 694 subcooled, xxiii, 25 supercritical, 24, 376 superficial basis, 699 superheated, xxiii, 25 superheater, 71, 201 875 876 Index surface fraction, 510 surfactant, 691 sweep gas, 387 system, 15 closed, 16 isolated, 16 open, 16 point, 713 simple, 131, 226 T temperature, 6, 254 absolute, 7 bubble, See bubble-point temperature dew, See dew-point temperature gradient, 44 reduced, 253 reference, 63, 115 See also reference state state variable, 18 thermal efficiency, xxiii, 97 thermodynamic efficiency, xxiii third law of thermodynamics, 33, 63, 728 three-root region, 264 throttle, xxiii, 68, 74, 162 tie line, 23, 371, 551 ton of refrigeration capacity, 209 total condenser, 101 trays, 102 triple product rule, 231, 246 true concentrations, 699 true mole fraction, 772 true moles, 792, 793 turbine, 72, 163, 166 multistage, 172 two-fluid theory, 504 T-x-y diagram, 370 U UCST, 552 UFL, 390 UNIFAC, 500, 514, 529, 562, 812 UNIQUAC, 509, 529, 812 unstable, xxiii, 268 unsteady-state, 74 unsymmetric convention activity coefficients, 446 upper critical solution temperature, 550 upper flammability limit, 390 V valve, 68, 162 See also throttle van der Waals, 253, 261, 469 area, 511 equation, See equation of state loop, 352 volume, 511 van Laar, 468, 470, 471, 474, 476, 812 van’t Hoff, 469, 653, 746 van’t Hoff equation, 653 shortcut, 655 vapor pressure, 24, 62, 254, 342, 353, 357, 376, 400 shortcut, 339 See also Antoine equation vector, 20 velocity gradient, 44 virial coefficient See equation of state, virial viscosity, 43 viscous dissipation, 43 VLE, xxiii, 372 VLLE, 434, 540, 548, 550 VOC, 387 volatile organic compounds, 387 voltage, 727 volume saturated liquid, 351 van der Waals, 511 volume fraction, 472, 510 volume of mixing, 105, 106 W wax, 564, 813 weak acid, 704 weak base, 704 weak electrolyte, 694 wet steam, xxiii, 24 Wilson equation, 505, 513 work, 15 expansion/contraction, 40 flow, 41 maximum, 164 minimum, 164 optimum, 158, 177 pump, 55, 202 shaft, 41, 54, 214 sign convention, 40 X x-y plot, 376, 419 Z Z, compressibility factor, 253 zeroth law of thermodynamics, 7 zwitterion, 721